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Properties

Label 2-354-177.176-c3-0-45
Degree $2$
Conductor $354$
Sign $0.954 + 0.298i$
Analytic cond. $20.8866$
Root an. cond. $4.57019$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + (2.99 + 4.24i)3-s + 4·4-s − 19.3i·5-s + (5.98 + 8.49i)6-s + 6.85·7-s + 8·8-s + (−9.10 + 25.4i)9-s − 38.6i·10-s + 4.18·11-s + (11.9 + 16.9i)12-s + 24.0i·13-s + 13.7·14-s + (82.0 − 57.7i)15-s + 16·16-s − 81.9i·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.575 + 0.817i)3-s + 0.5·4-s − 1.72i·5-s + (0.407 + 0.578i)6-s + 0.370·7-s + 0.353·8-s + (−0.337 + 0.941i)9-s − 1.22i·10-s + 0.114·11-s + (0.287 + 0.408i)12-s + 0.513i·13-s + 0.261·14-s + (1.41 − 0.994i)15-s + 0.250·16-s − 1.16i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.954 + 0.298i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.954 + 0.298i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(354\)    =    \(2 \cdot 3 \cdot 59\)
Sign: $0.954 + 0.298i$
Analytic conductor: \(20.8866\)
Root analytic conductor: \(4.57019\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{354} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 354,\ (\ :3/2),\ 0.954 + 0.298i)\)

Particular Values

\(L(2)\) \(\approx\) \(3.815162451\)
\(L(\frac12)\) \(\approx\) \(3.815162451\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2T \)
3 \( 1 + (-2.99 - 4.24i)T \)
59 \( 1 + (359. - 275. i)T \)
good5 \( 1 + 19.3iT - 125T^{2} \)
7 \( 1 - 6.85T + 343T^{2} \)
11 \( 1 - 4.18T + 1.33e3T^{2} \)
13 \( 1 - 24.0iT - 2.19e3T^{2} \)
17 \( 1 + 81.9iT - 4.91e3T^{2} \)
19 \( 1 - 122.T + 6.85e3T^{2} \)
23 \( 1 - 216.T + 1.21e4T^{2} \)
29 \( 1 + 136. iT - 2.43e4T^{2} \)
31 \( 1 + 342. iT - 2.97e4T^{2} \)
37 \( 1 - 346. iT - 5.06e4T^{2} \)
41 \( 1 + 280. iT - 6.89e4T^{2} \)
43 \( 1 - 353. iT - 7.95e4T^{2} \)
47 \( 1 - 88.0T + 1.03e5T^{2} \)
53 \( 1 + 89.3iT - 1.48e5T^{2} \)
61 \( 1 - 198. iT - 2.26e5T^{2} \)
67 \( 1 - 856. iT - 3.00e5T^{2} \)
71 \( 1 - 384. iT - 3.57e5T^{2} \)
73 \( 1 - 586. iT - 3.89e5T^{2} \)
79 \( 1 - 417.T + 4.93e5T^{2} \)
83 \( 1 + 1.26e3T + 5.71e5T^{2} \)
89 \( 1 + 329.T + 7.04e5T^{2} \)
97 \( 1 + 49.9iT - 9.12e5T^{2} \)
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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

−11.40668787564304834157253849546, −9.739325422996193767932737412840, −9.277839841461347920528404318849, −8.305433981733105489090885963236, −7.35649483970720898498243719292, −5.55653638084733411346933487341, −4.87599046479367162209331282085, −4.20871397556555032042599090907, −2.78452039912301416543641817187, −1.14996530646586515855078318920, 1.56435630710443022463894070152, 2.99835476824307430502176278717, 3.42177879966936297514681729423, 5.32211892128060491311790603137, 6.50406886843729787743713533044, 7.12440150897679320170718683369, 7.87998873099009699841182856623, 9.172037350083221752325851114786, 10.57609423826726052105056525061, 11.05101638169660704777035109305

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