Properties

Label 2-354-177.5-c2-0-32
Degree $2$
Conductor $354$
Sign $0.143 + 0.989i$
Analytic cond. $9.64580$
Root an. cond. $3.10576$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.451 + 1.34i)2-s + (2.35 − 1.86i)3-s + (−1.59 − 1.21i)4-s + (−1.37 − 0.547i)5-s + (1.43 + 3.99i)6-s + (−4.55 + 2.10i)7-s + (2.34 − 1.58i)8-s + (2.07 − 8.75i)9-s + (1.35 − 1.59i)10-s + (−4.49 − 1.24i)11-s + (−5.99 + 0.114i)12-s + (3.32 − 6.28i)13-s + (−0.767 − 7.06i)14-s + (−4.25 + 1.26i)15-s + (1.07 + 3.85i)16-s + (13.6 − 29.4i)17-s + ⋯
L(s)  = 1  + (−0.225 + 0.670i)2-s + (0.784 − 0.620i)3-s + (−0.398 − 0.302i)4-s + (−0.274 − 0.109i)5-s + (0.238 + 0.665i)6-s + (−0.651 + 0.301i)7-s + (0.292 − 0.198i)8-s + (0.230 − 0.973i)9-s + (0.135 − 0.159i)10-s + (−0.409 − 0.113i)11-s + (−0.499 + 0.00957i)12-s + (0.256 − 0.483i)13-s + (−0.0548 − 0.504i)14-s + (−0.283 + 0.0845i)15-s + (0.0668 + 0.240i)16-s + (0.801 − 1.73i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.143 + 0.989i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.143 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(354\)    =    \(2 \cdot 3 \cdot 59\)
Sign: $0.143 + 0.989i$
Analytic conductor: \(9.64580\)
Root analytic conductor: \(3.10576\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{354} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 354,\ (\ :1),\ 0.143 + 0.989i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.931612 - 0.806325i\)
\(L(\frac12)\) \(\approx\) \(0.931612 - 0.806325i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.451 - 1.34i)T \)
3 \( 1 + (-2.35 + 1.86i)T \)
59 \( 1 + (58.6 + 6.17i)T \)
good5 \( 1 + (1.37 + 0.547i)T + (18.1 + 17.1i)T^{2} \)
7 \( 1 + (4.55 - 2.10i)T + (31.7 - 37.3i)T^{2} \)
11 \( 1 + (4.49 + 1.24i)T + (103. + 62.3i)T^{2} \)
13 \( 1 + (-3.32 + 6.28i)T + (-94.8 - 139. i)T^{2} \)
17 \( 1 + (-13.6 + 29.4i)T + (-187. - 220. i)T^{2} \)
19 \( 1 + (16.1 - 3.56i)T + (327. - 151. i)T^{2} \)
23 \( 1 + (-28.7 - 4.70i)T + (501. + 168. i)T^{2} \)
29 \( 1 + (10.9 + 32.6i)T + (-669. + 508. i)T^{2} \)
31 \( 1 + (57.4 + 12.6i)T + (872. + 403. i)T^{2} \)
37 \( 1 + (-17.3 + 25.6i)T + (-506. - 1.27e3i)T^{2} \)
41 \( 1 + (21.2 - 3.48i)T + (1.59e3 - 536. i)T^{2} \)
43 \( 1 + (-12.0 - 43.3i)T + (-1.58e3 + 953. i)T^{2} \)
47 \( 1 + (-58.4 + 23.2i)T + (1.60e3 - 1.51e3i)T^{2} \)
53 \( 1 + (16.4 - 13.9i)T + (454. - 2.77e3i)T^{2} \)
61 \( 1 + (-35.5 - 11.9i)T + (2.96e3 + 2.25e3i)T^{2} \)
67 \( 1 + (-50.2 - 74.1i)T + (-1.66e3 + 4.17e3i)T^{2} \)
71 \( 1 + (-124. + 49.4i)T + (3.65e3 - 3.46e3i)T^{2} \)
73 \( 1 + (-64.4 + 7.00i)T + (5.20e3 - 1.14e3i)T^{2} \)
79 \( 1 + (-121. + 72.8i)T + (2.92e3 - 5.51e3i)T^{2} \)
83 \( 1 + (58.7 - 3.18i)T + (6.84e3 - 744. i)T^{2} \)
89 \( 1 + (-40.3 - 119. i)T + (-6.30e3 + 4.79e3i)T^{2} \)
97 \( 1 + (19.4 + 2.11i)T + (9.18e3 + 2.02e3i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.00757738550039882181812390745, −9.620202157168119018453305709179, −9.176176415074760811093786511622, −7.999469441369437986276008562469, −7.46222336205643183037914662279, −6.39654970377838080261394373847, −5.36127314305321563752301228453, −3.76615149005439512523250986597, −2.54582707197823174470237190588, −0.54633424450191005137059947164, 1.84539825303715925050128712229, 3.35069137690063772746933834176, 3.92254309752404297572621733514, 5.30796555186743661723409933862, 6.91604293959541336176088284622, 7.994781701657561668704714876357, 8.867125790825336148948067737493, 9.626163617159308461552726684219, 10.67261354404262220233558404631, 10.96246037521641859763918800025

Graph of the $Z$-function along the critical line