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Properties

Label 2-357-119.16-c1-0-2
Degree $2$
Conductor $357$
Sign $-0.0665 - 0.997i$
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  + (−1.34 + 2.32i)2-s + (0.866 − 0.5i)3-s + (−2.61 − 4.52i)4-s + (−2.98 − 1.72i)5-s + 2.68i·6-s + (0.462 + 2.60i)7-s + 8.66·8-s + (0.499 − 0.866i)9-s + (8.01 − 4.62i)10-s + (−0.636 + 0.367i)11-s + (−4.52 − 2.61i)12-s + 6.64·13-s + (−6.68 − 2.42i)14-s − 3.44·15-s + (−6.41 + 11.1i)16-s + (1.41 + 3.87i)17-s + ⋯
L(s)  = 1  + (−0.950 + 1.64i)2-s + (0.499 − 0.288i)3-s + (−1.30 − 2.26i)4-s + (−1.33 − 0.770i)5-s + 1.09i·6-s + (0.174 + 0.984i)7-s + 3.06·8-s + (0.166 − 0.288i)9-s + (2.53 − 1.46i)10-s + (−0.191 + 0.110i)11-s + (−1.30 − 0.753i)12-s + 1.84·13-s + (−1.78 − 0.647i)14-s − 0.889·15-s + (−1.60 + 2.77i)16-s + (0.343 + 0.939i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 357 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0665 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 357 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0665 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(357\)    =    \(3 \cdot 7 \cdot 17\)
Sign: $-0.0665 - 0.997i$
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: \((2,\ 357,\ (\ :1/2),\ -0.0665 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.525984 + 0.562251i\)
\(L(\frac12)\) \(\approx\) \(0.525984 + 0.562251i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.462 - 2.60i)T \)
17 \( 1 + (-1.41 - 3.87i)T \)
good2 \( 1 + (1.34 - 2.32i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (2.98 + 1.72i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.636 - 0.367i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 6.64T + 13T^{2} \)
19 \( 1 + (-0.832 + 1.44i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.08 - 2.35i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 4.36iT - 29T^{2} \)
31 \( 1 + (-2.56 + 1.48i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.711 - 0.410i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 7.21iT - 41T^{2} \)
43 \( 1 - 3.15T + 43T^{2} \)
47 \( 1 + (-2.84 + 4.93i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.253 - 0.438i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.46 - 9.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7.47 - 4.31i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.05 - 1.82i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 13.8iT - 71T^{2} \)
73 \( 1 + (1.85 - 1.07i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (10.0 + 5.81i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 3.50T + 83T^{2} \)
89 \( 1 + (-0.872 + 1.51i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 12.5iT - 97T^{2} \)
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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

−11.61364757196199910191136316988, −10.52642534015192083044623596543, −8.986303418637065948020404720662, −8.753017848382538751552543186627, −8.081751746797190063424464063136, −7.27891750642885491931912887243, −6.11149559224833247240817185078, −5.17824381715527722460386849533, −3.85600489344656477066222737603, −1.15984293016658998155885651801, 0.940921945906383786902859938619, 2.91360139578465502831441475850, 3.64056217698845053690819606026, 4.37397340705533388600854048448, 7.01628281140329115214383986369, 7.977660649895566526067398986295, 8.426739404840137574757821610506, 9.632333621903666567414563663836, 10.49549236303609415885577653643, 11.22713412023867629205784080181

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