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Properties

Label 2-354-177.176-c5-0-0
Degree $2$
Conductor $354$
Sign $-0.365 - 0.930i$
Analytic cond. $56.7758$
Root an. cond. $7.53497$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s + (15.3 − 2.47i)3-s + 16·4-s − 109. i·5-s + (−61.5 + 9.91i)6-s + 18.6·7-s − 64·8-s + (230. − 76.2i)9-s + 438. i·10-s − 492.·11-s + (246. − 39.6i)12-s + 588. i·13-s − 74.6·14-s + (−271. − 1.68e3i)15-s + 256·16-s + 1.24e3i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.987 − 0.158i)3-s + 0.5·4-s − 1.96i·5-s + (−0.698 + 0.112i)6-s + 0.144·7-s − 0.353·8-s + (0.949 − 0.313i)9-s + 1.38i·10-s − 1.22·11-s + (0.493 − 0.0794i)12-s + 0.965i·13-s − 0.101·14-s + (−0.312 − 1.93i)15-s + 0.250·16-s + 1.04i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.365 - 0.930i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.365 - 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(354\)    =    \(2 \cdot 3 \cdot 59\)
Sign: $-0.365 - 0.930i$
Analytic conductor: \(56.7758\)
Root analytic conductor: \(7.53497\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{354} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 354,\ (\ :5/2),\ -0.365 - 0.930i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.1662102029\)
\(L(\frac12)\) \(\approx\) \(0.1662102029\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 4T \)
3 \( 1 + (-15.3 + 2.47i)T \)
59 \( 1 + (-5.68e3 - 2.61e4i)T \)
good5 \( 1 + 109. iT - 3.12e3T^{2} \)
7 \( 1 - 18.6T + 1.68e4T^{2} \)
11 \( 1 + 492.T + 1.61e5T^{2} \)
13 \( 1 - 588. iT - 3.71e5T^{2} \)
17 \( 1 - 1.24e3iT - 1.41e6T^{2} \)
19 \( 1 + 1.07e3T + 2.47e6T^{2} \)
23 \( 1 + 3.96e3T + 6.43e6T^{2} \)
29 \( 1 - 4.19e3iT - 2.05e7T^{2} \)
31 \( 1 - 2.17e3iT - 2.86e7T^{2} \)
37 \( 1 + 3.28e3iT - 6.93e7T^{2} \)
41 \( 1 - 2.10e3iT - 1.15e8T^{2} \)
43 \( 1 - 1.71e4iT - 1.47e8T^{2} \)
47 \( 1 - 2.06e4T + 2.29e8T^{2} \)
53 \( 1 + 6.82e3iT - 4.18e8T^{2} \)
61 \( 1 + 2.76e4iT - 8.44e8T^{2} \)
67 \( 1 + 4.61e4iT - 1.35e9T^{2} \)
71 \( 1 + 7.02e4iT - 1.80e9T^{2} \)
73 \( 1 - 1.96e4iT - 2.07e9T^{2} \)
79 \( 1 - 4.42e4T + 3.07e9T^{2} \)
83 \( 1 + 3.81e4T + 3.93e9T^{2} \)
89 \( 1 - 4.48e4T + 5.58e9T^{2} \)
97 \( 1 + 3.00e4iT - 8.58e9T^{2} \)
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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

−10.64316260994788901676468115118, −9.655527474030530696741222871859, −8.935683846115144764569356614077, −8.226786181227032663707244117754, −7.76374250842917962382909185225, −6.20840896044678485008283952139, −4.88278025527096003717824628845, −3.92160605368196370832103113631, −2.12897266315810870716011522577, −1.39496033506535856475252355962, 0.04280094143040329623675696181, 2.31605571548781997156556545512, 2.69285371188246849751992936727, 3.85891200192260052266256054638, 5.71567353471657265125206425954, 6.93034491563958652169497011087, 7.68776932350167680740575980798, 8.226724752524326850870676101003, 9.708346834036560924806951449744, 10.28434182536904623510570151554

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