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Properties

Label 2-357-7.2-c1-0-3
Degree $2$
Conductor $357$
Sign $-0.614 - 0.788i$
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  + (−1.18 + 2.05i)2-s + (−0.5 − 0.866i)3-s + (−1.81 − 3.13i)4-s + (−0.740 + 1.28i)5-s + 2.37·6-s + (2.18 − 1.49i)7-s + 3.85·8-s + (−0.499 + 0.866i)9-s + (−1.75 − 3.04i)10-s + (1.18 + 2.05i)11-s + (−1.81 + 3.13i)12-s + 3.22·13-s + (0.469 + 6.25i)14-s + 1.48·15-s + (−0.944 + 1.63i)16-s + (0.5 + 0.866i)17-s + ⋯
L(s)  = 1  + (−0.838 + 1.45i)2-s + (−0.288 − 0.499i)3-s + (−0.906 − 1.56i)4-s + (−0.331 + 0.573i)5-s + 0.968·6-s + (0.826 − 0.563i)7-s + 1.36·8-s + (−0.166 + 0.288i)9-s + (−0.555 − 0.962i)10-s + (0.357 + 0.619i)11-s + (−0.523 + 0.906i)12-s + 0.895·13-s + (0.125 + 1.67i)14-s + 0.382·15-s + (−0.236 + 0.409i)16-s + (0.121 + 0.210i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 357 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.614 - 0.788i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 357 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.614 - 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(357\)    =    \(3 \cdot 7 \cdot 17\)
Sign: $-0.614 - 0.788i$
Analytic conductor: \(2.85065\)
Root analytic conductor: \(1.68838\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{357} (205, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 357,\ (\ :1/2),\ -0.614 - 0.788i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.319260 + 0.653479i\)
\(L(\frac12)\) \(\approx\) \(0.319260 + 0.653479i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (-2.18 + 1.49i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
good2 \( 1 + (1.18 - 2.05i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (0.740 - 1.28i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.18 - 2.05i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 3.22T + 13T^{2} \)
19 \( 1 + (3.05 - 5.28i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.92 - 3.33i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 2.08T + 29T^{2} \)
31 \( 1 + (-1.11 - 1.93i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.87 - 3.24i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 9.80T + 41T^{2} \)
43 \( 1 + 4.55T + 43T^{2} \)
47 \( 1 + (-0.867 + 1.50i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.06 - 7.04i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.68 + 9.85i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.65 + 6.33i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.44 - 12.9i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 5.87T + 71T^{2} \)
73 \( 1 + (-5.56 - 9.63i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.83 - 6.64i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 3.97T + 83T^{2} \)
89 \( 1 + (-2.19 + 3.79i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 12.4T + 97T^{2} \)
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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

−11.55273320664263927779605137463, −10.71498929633613864552098648688, −9.797770884756152730380307517951, −8.540967894638894757107614086033, −7.88059274533604757231530437580, −7.14795530480445676998593029969, −6.36412872375964516260366682592, −5.38030862825009342053722633210, −3.94649660806114892640753499280, −1.44321388345946621322416365088, 0.78185345525297621580742082947, 2.37966653603208147607110501002, 3.80255612589950436935239634392, 4.76313918271292142923204668836, 6.15528551224652407691585324321, 7.998269313527799663319765042263, 8.792135779660358517762446546042, 9.164208797460418355657091950571, 10.50034013884421935969255024708, 11.09422007938116410373083058906

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