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Properties

Label 2-354-177.176-c3-0-57
Degree $2$
Conductor $354$
Sign $-0.917 + 0.397i$
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 + (−3.62 + 3.71i)3-s + 4·4-s − 18.5i·5-s + (−7.25 + 7.43i)6-s + 9.26·7-s + 8·8-s + (−0.657 − 26.9i)9-s − 37.1i·10-s − 54.2·11-s + (−14.5 + 14.8i)12-s + 34.8i·13-s + 18.5·14-s + (69.0 + 67.3i)15-s + 16·16-s + 8.98i·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.698 + 0.715i)3-s + 0.5·4-s − 1.66i·5-s + (−0.493 + 0.506i)6-s + 0.500·7-s + 0.353·8-s + (−0.0243 − 0.999i)9-s − 1.17i·10-s − 1.48·11-s + (−0.349 + 0.357i)12-s + 0.744i·13-s + 0.353·14-s + (1.18 + 1.15i)15-s + 0.250·16-s + 0.128i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.917 + 0.397i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.917 + 0.397i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(354\)    =    \(2 \cdot 3 \cdot 59\)
Sign: $-0.917 + 0.397i$
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: \((2,\ 354,\ (\ :3/2),\ -0.917 + 0.397i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.6358287879\)
\(L(\frac12)\) \(\approx\) \(0.6358287879\)
\(L(\frac{5}{2})\) 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 \( 1 + (3.62 - 3.71i)T \)
59 \( 1 + (419. + 171. i)T \)
good5 \( 1 + 18.5iT - 125T^{2} \)
7 \( 1 - 9.26T + 343T^{2} \)
11 \( 1 + 54.2T + 1.33e3T^{2} \)
13 \( 1 - 34.8iT - 2.19e3T^{2} \)
17 \( 1 - 8.98iT - 4.91e3T^{2} \)
19 \( 1 + 123.T + 6.85e3T^{2} \)
23 \( 1 + 94.6T + 1.21e4T^{2} \)
29 \( 1 - 57.9iT - 2.43e4T^{2} \)
31 \( 1 + 8.78iT - 2.97e4T^{2} \)
37 \( 1 + 124. iT - 5.06e4T^{2} \)
41 \( 1 + 316. iT - 6.89e4T^{2} \)
43 \( 1 - 249. iT - 7.95e4T^{2} \)
47 \( 1 + 143.T + 1.03e5T^{2} \)
53 \( 1 + 716. iT - 1.48e5T^{2} \)
61 \( 1 + 583. iT - 2.26e5T^{2} \)
67 \( 1 - 11.7iT - 3.00e5T^{2} \)
71 \( 1 - 14.4iT - 3.57e5T^{2} \)
73 \( 1 - 1.14e3iT - 3.89e5T^{2} \)
79 \( 1 + 491.T + 4.93e5T^{2} \)
83 \( 1 - 1.08e3T + 5.71e5T^{2} \)
89 \( 1 - 1.31e3T + 7.04e5T^{2} \)
97 \( 1 + 823. iT - 9.12e5T^{2} \)
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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

−10.81642561116383795213822655736, −9.859372537905312166882473554998, −8.757207925576503436465811378471, −7.949438021398997439876292255192, −6.36139704653046861156124258684, −5.29706966132748973668647801597, −4.79107048521284804862194718210, −3.94036034095285143971656500808, −1.92196012142223772390397200114, −0.17351350221342067026362347043, 2.12112814548887811459164466208, 2.97300053393559329033107897819, 4.63059494033172339799021344848, 5.81177002969579748236446691305, 6.45860670289456126494281335237, 7.54867665357199830485605484358, 8.037508693193600968872294387534, 10.41133397739874423841922096417, 10.54263889952573944723859555537, 11.40839048984207070604236230489

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