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Properties

Label 2-351-39.32-c1-0-5
Degree $2$
Conductor $351$
Sign $0.194 - 0.980i$
Analytic cond. $2.80274$
Root an. cond. $1.67414$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.147 − 0.0394i)2-s + (−1.71 + 0.988i)4-s + (0.671 + 0.671i)5-s + (0.383 − 1.43i)7-s + (−0.428 + 0.428i)8-s + (0.125 + 0.0723i)10-s + (1.27 + 4.77i)11-s + (−0.563 + 3.56i)13-s − 0.225i·14-s + (1.93 − 3.34i)16-s + (3.08 + 5.35i)17-s + (−1.68 − 0.451i)19-s + (−1.81 − 0.485i)20-s + (0.376 + 0.652i)22-s + (−3.74 + 6.49i)23-s + ⋯
L(s)  = 1  + (0.104 − 0.0278i)2-s + (−0.855 + 0.494i)4-s + (0.300 + 0.300i)5-s + (0.144 − 0.540i)7-s + (−0.151 + 0.151i)8-s + (0.0396 + 0.0228i)10-s + (0.385 + 1.43i)11-s + (−0.156 + 0.987i)13-s − 0.0603i·14-s + (0.482 − 0.835i)16-s + (0.749 + 1.29i)17-s + (−0.386 − 0.103i)19-s + (−0.405 − 0.108i)20-s + (0.0803 + 0.139i)22-s + (−0.781 + 1.35i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 351 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.194 - 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 351 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.194 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(351\)    =    \(3^{3} \cdot 13\)
Sign: $0.194 - 0.980i$
Analytic conductor: \(2.80274\)
Root analytic conductor: \(1.67414\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{351} (188, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 351,\ (\ :1/2),\ 0.194 - 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.878338 + 0.721449i\)
\(L(\frac12)\) \(\approx\) \(0.878338 + 0.721449i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 + (0.563 - 3.56i)T \)
good2 \( 1 + (-0.147 + 0.0394i)T + (1.73 - i)T^{2} \)
5 \( 1 + (-0.671 - 0.671i)T + 5iT^{2} \)
7 \( 1 + (-0.383 + 1.43i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (-1.27 - 4.77i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (-3.08 - 5.35i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.68 + 0.451i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (3.74 - 6.49i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.18 + 0.681i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.06 - 2.06i)T - 31iT^{2} \)
37 \( 1 + (1.85 - 0.497i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-9.12 + 2.44i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-6.31 + 3.64i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.42 + 5.42i)T - 47iT^{2} \)
53 \( 1 + 8.05iT - 53T^{2} \)
59 \( 1 + (4.49 + 1.20i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (1.54 + 2.67i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.516 + 1.92i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (0.385 - 1.43i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (7.05 + 7.05i)T + 73iT^{2} \)
79 \( 1 + 4.39T + 79T^{2} \)
83 \( 1 + (6.90 + 6.90i)T + 83iT^{2} \)
89 \( 1 + (-2.33 - 8.70i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-12.5 - 3.35i)T + (84.0 + 48.5i)T^{2} \)
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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

−11.97657128824486910828645159562, −10.59452454135947790143431278012, −9.794213392634675771061677316100, −9.038161002678372508312061999266, −7.84353910266402746065500778603, −7.07913651975373699630646399566, −5.78883582712398389641713794068, −4.42082147972383250278610821701, −3.80766849935747611306102626558, −1.92516540685930323722544721102, 0.828392325676112274718163783978, 2.91121055692667176973755054037, 4.33357131093403495345536256170, 5.59660083755541495862845078086, 5.92296825187062217536034390199, 7.68131752483138749870615901691, 8.720846884736681549618852000588, 9.273517686582285035407963734032, 10.30201638234986989710863946368, 11.20837672586588466536625550527

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