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Properties

Label 2-14e2-196.111-c1-0-23
Degree 22
Conductor 196196
Sign −0.805−0.592i-0.805 - 0.592i
Analytic cond. 1.565061.56506
Root an. cond. 1.251021.25102
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.25 − 0.653i)2-s + (−0.635 − 0.797i)3-s + (1.14 + 1.63i)4-s + (−0.726 + 0.579i)5-s + (0.276 + 1.41i)6-s + (−2.60 + 0.472i)7-s + (−0.365 − 2.80i)8-s + (0.436 − 1.91i)9-s + (1.28 − 0.251i)10-s + (−5.04 + 1.15i)11-s + (0.578 − 1.95i)12-s + (−2.71 + 0.618i)13-s + (3.57 + 1.10i)14-s + (0.923 + 0.210i)15-s + (−1.37 + 3.75i)16-s + (−2.67 + 5.55i)17-s + ⋯
L(s)  = 1  + (−0.886 − 0.462i)2-s + (−0.367 − 0.460i)3-s + (0.572 + 0.819i)4-s + (−0.324 + 0.259i)5-s + (0.112 + 0.577i)6-s + (−0.983 + 0.178i)7-s + (−0.129 − 0.991i)8-s + (0.145 − 0.637i)9-s + (0.407 − 0.0796i)10-s + (−1.52 + 0.347i)11-s + (0.166 − 0.564i)12-s + (−0.751 + 0.171i)13-s + (0.955 + 0.296i)14-s + (0.238 + 0.0544i)15-s + (−0.343 + 0.939i)16-s + (−0.648 + 1.34i)17-s + ⋯

Functional equation

Λ(s)=(196s/2 ΓC(s) L(s)=((−0.805−0.592i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.805 - 0.592i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(196s/2 ΓC(s+1/2) L(s)=((−0.805−0.592i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.805 - 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 196196    =    22⋅722^{2} \cdot 7^{2}
Sign: −0.805−0.592i-0.805 - 0.592i
Analytic conductor: 1.565061.56506
Root analytic conductor: 1.251021.25102
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ196(111,⋅)\chi_{196} (111, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 196, ( :1/2), −0.805−0.592i)(2,\ 196,\ (\ :1/2),\ -0.805 - 0.592i)

Particular Values

L(1)L(1) ≈\approx 0.00178133+0.00543040i0.00178133 + 0.00543040i
L(12)L(\frac12) ≈\approx 0.00178133+0.00543040i0.00178133 + 0.00543040i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1+(1.25+0.653i)T 1 + (1.25 + 0.653i)T
7 1+(2.60−0.472i)T 1 + (2.60 - 0.472i)T
good3 1+(0.635+0.797i)T+(−0.667+2.92i)T2 1 + (0.635 + 0.797i)T + (-0.667 + 2.92i)T^{2}
5 1+(0.726−0.579i)T+(1.11−4.87i)T2 1 + (0.726 - 0.579i)T + (1.11 - 4.87i)T^{2}
11 1+(5.04−1.15i)T+(9.91−4.77i)T2 1 + (5.04 - 1.15i)T + (9.91 - 4.77i)T^{2}
13 1+(2.71−0.618i)T+(11.7−5.64i)T2 1 + (2.71 - 0.618i)T + (11.7 - 5.64i)T^{2}
17 1+(2.67−5.55i)T+(−10.5−13.2i)T2 1 + (2.67 - 5.55i)T + (-10.5 - 13.2i)T^{2}
19 1−5.31T+19T2 1 - 5.31T + 19T^{2}
23 1+(2.67+5.54i)T+(−14.3+17.9i)T2 1 + (2.67 + 5.54i)T + (-14.3 + 17.9i)T^{2}
29 1+(6.99+3.37i)T+(18.0+22.6i)T2 1 + (6.99 + 3.37i)T + (18.0 + 22.6i)T^{2}
31 1−2.80T+31T2 1 - 2.80T + 31T^{2}
37 1+(5.03+2.42i)T+(23.0+28.9i)T2 1 + (5.03 + 2.42i)T + (23.0 + 28.9i)T^{2}
41 1+(−8.64+6.89i)T+(9.12−39.9i)T2 1 + (-8.64 + 6.89i)T + (9.12 - 39.9i)T^{2}
43 1+(−1.76−1.40i)T+(9.56+41.9i)T2 1 + (-1.76 - 1.40i)T + (9.56 + 41.9i)T^{2}
47 1+(1.31+5.75i)T+(−42.3+20.3i)T2 1 + (1.31 + 5.75i)T + (-42.3 + 20.3i)T^{2}
53 1+(−3.53+1.70i)T+(33.0−41.4i)T2 1 + (-3.53 + 1.70i)T + (33.0 - 41.4i)T^{2}
59 1+(3.63−4.55i)T+(−13.1−57.5i)T2 1 + (3.63 - 4.55i)T + (-13.1 - 57.5i)T^{2}
61 1+(2.83−5.89i)T+(−38.0−47.6i)T2 1 + (2.83 - 5.89i)T + (-38.0 - 47.6i)T^{2}
67 1+7.05iT−67T2 1 + 7.05iT - 67T^{2}
71 1+(0.607+1.26i)T+(−44.2+55.5i)T2 1 + (0.607 + 1.26i)T + (-44.2 + 55.5i)T^{2}
73 1+(4.23+0.966i)T+(65.7+31.6i)T2 1 + (4.23 + 0.966i)T + (65.7 + 31.6i)T^{2}
79 1−1.47iT−79T2 1 - 1.47iT - 79T^{2}
83 1+(2.36−10.3i)T+(−74.7−36.0i)T2 1 + (2.36 - 10.3i)T + (-74.7 - 36.0i)T^{2}
89 1+(−8.05−1.83i)T+(80.1+38.6i)T2 1 + (-8.05 - 1.83i)T + (80.1 + 38.6i)T^{2}
97 1+11.4iT−97T2 1 + 11.4iT - 97T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−11.99202616810860235223781476753, −10.80353041406960920839485840292, −10.00016211437389784676564425758, −9.084428460002264563367327170920, −7.72867592656735619927526186785, −7.02876643218892538618495577477, −5.86244877506908496735719288944, −3.76886022862283911711103673017, −2.37488739158837668192039402678, −0.00634243796433922256916029322, 2.76497001027624577661157140201, 4.89781351480617154403715533954, 5.72802561074281026782542938933, 7.30789745538639549822357913332, 7.84674337712099212610441053750, 9.365081809400240811153777099175, 9.968338351648493896071417921879, 10.87671885031570155697094803908, 11.79919690987313018883990690366, 13.16718862020376574403056120678

Graph of the ZZ-function along the critical line