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Properties

Label 2-357-119.67-c1-0-10
Degree $2$
Conductor $357$
Sign $0.655 - 0.755i$
Analytic cond. $2.85065$
Root an. cond. $1.68838$
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  + (0.262 + 0.454i)2-s + (0.866 + 0.5i)3-s + (0.862 − 1.49i)4-s + (−3.41 + 1.97i)5-s + 0.524i·6-s + (2.63 + 0.177i)7-s + 1.95·8-s + (0.499 + 0.866i)9-s + (−1.79 − 1.03i)10-s + (2.72 + 1.57i)11-s + (1.49 − 0.862i)12-s + 0.881·13-s + (0.611 + 1.24i)14-s − 3.94·15-s + (−1.21 − 2.09i)16-s + (1.84 + 3.68i)17-s + ⋯
L(s)  = 1  + (0.185 + 0.321i)2-s + (0.499 + 0.288i)3-s + (0.431 − 0.746i)4-s + (−1.52 + 0.882i)5-s + 0.214i·6-s + (0.997 + 0.0671i)7-s + 0.691·8-s + (0.166 + 0.288i)9-s + (−0.567 − 0.327i)10-s + (0.821 + 0.474i)11-s + (0.431 − 0.248i)12-s + 0.244·13-s + (0.163 + 0.333i)14-s − 1.01·15-s + (−0.302 − 0.524i)16-s + (0.446 + 0.894i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 357 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.655 - 0.755i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 357 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.655 - 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(357\)    =    \(3 \cdot 7 \cdot 17\)
Sign: $0.655 - 0.755i$
Analytic conductor: \(2.85065\)
Root analytic conductor: \(1.68838\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{357} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 357,\ (\ :1/2),\ 0.655 - 0.755i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.58324 + 0.722374i\)
\(L(\frac12)\) \(\approx\) \(1.58324 + 0.722374i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.866 - 0.5i)T \)
7 \( 1 + (-2.63 - 0.177i)T \)
17 \( 1 + (-1.84 - 3.68i)T \)
good2 \( 1 + (-0.262 - 0.454i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (3.41 - 1.97i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.72 - 1.57i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 0.881T + 13T^{2} \)
19 \( 1 + (-1.76 - 3.06i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.87 - 1.65i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 8.73iT - 29T^{2} \)
31 \( 1 + (3.05 + 1.76i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (10.3 - 5.98i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 6.52iT - 41T^{2} \)
43 \( 1 - 3.56T + 43T^{2} \)
47 \( 1 + (4.34 + 7.53i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.38 + 4.13i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.39 - 11.0i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.80 + 3.92i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.65 - 2.86i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 4.10iT - 71T^{2} \)
73 \( 1 + (9.52 + 5.50i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.03 + 1.75i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 10.1T + 83T^{2} \)
89 \( 1 + (-0.556 - 0.964i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 4.98iT - 97T^{2} \)
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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

−11.63048154706598337402200069172, −10.66511660919621862005236586458, −9.999199500881568120275617436161, −8.496897790982751649126543465504, −7.71638101953854417895571573475, −7.02370080689918151665436582900, −5.79116522903046160132448761868, −4.39105369061159656652981458181, −3.62227609161636817200007119401, −1.83957328741200693843299304103, 1.33320166115611011257557441541, 3.18320262830225977610534022076, 4.05136471545100269061000800570, 5.01603397638725231463547044876, 7.02648230698524251511873424166, 7.64466203904393557977710694208, 8.470623173128621758527164273794, 9.023699786590689240525482207235, 10.93298463540694205015564011289, 11.51003780527628155021779919612

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