↑

Properties

Label 2-14e2-196.111-c1-0-23
Degree $2$
Conductor $196$
Sign $-0.805 - 0.592i$
Analytic cond. $1.56506$
Root an. cond. $1.25102$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

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

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

Invariants

Degree: \(2\)
Conductor: \(196\)    =    \(2^{2} \cdot 7^{2}\)
Sign: $-0.805 - 0.592i$
Analytic conductor: \(1.56506\)
Root analytic conductor: \(1.25102\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{196} (111, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 196,\ (\ :1/2),\ -0.805 - 0.592i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00178133 + 0.00543040i\)
\(L(\frac12)\) \(\approx\) \(0.00178133 + 0.00543040i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

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

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 $Z$-function along the critical line