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Properties

Label 2-357-119.16-c1-0-4
Degree $2$
Conductor $357$
Sign $0.438 - 0.898i$
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 + (−4.11 − 0.249i)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 + (−0.998 − 0.0604i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 357 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.438 - 0.898i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 357 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.438 - 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(357\)    =    \(3 \cdot 7 \cdot 17\)
Sign: $0.438 - 0.898i$
Analytic conductor: \(2.85065\)
Root analytic conductor: \(1.68838\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{357} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 357,\ (\ :1/2),\ 0.438 - 0.898i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.32802 + 0.830102i\)
\(L(\frac12)\) \(\approx\) \(1.32802 + 0.830102i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.11 + 0.249i)T \)
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

−11.39806902508059341580747367300, −10.77470760787167096065064117669, −9.898974145476101123517083137181, −9.272065556694997867490369559451, −7.60330735587143460814189231796, −6.58973053435133730207438212046, −6.04109694086386030522769282246, −4.62360908018525155361131728578, −3.05058854721944335046592540054, −2.31795595338319772106009839941, 1.14589755894535232900192843275, 2.55098698196461976058408579278, 4.77436044571016195388496470859, 5.69245261581361752302719703077, 6.16720643921164112156300832824, 7.09695701314953414220559691932, 8.653870980198854991499687476712, 9.611784541669672929939161030524, 10.32912588969409012327097543246, 11.07328062641132774310198509776

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