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Properties

Label 2-357-119.67-c1-0-6
Degree $2$
Conductor $357$
Sign $0.264 - 0.964i$
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  + (0.667 + 1.15i)2-s + (−0.866 − 0.5i)3-s + (0.109 − 0.190i)4-s + (−0.786 + 0.453i)5-s − 1.33i·6-s + (−1.35 + 2.26i)7-s + 2.96·8-s + (0.499 + 0.866i)9-s + (−1.04 − 0.605i)10-s + (4.27 + 2.46i)11-s + (−0.190 + 0.109i)12-s + 1.97·13-s + (−3.52 − 0.0564i)14-s + 0.907·15-s + (1.75 + 3.04i)16-s + (−0.772 + 4.05i)17-s + ⋯
L(s)  = 1  + (0.471 + 0.817i)2-s + (−0.499 − 0.288i)3-s + (0.0549 − 0.0950i)4-s + (−0.351 + 0.203i)5-s − 0.544i·6-s + (−0.513 + 0.857i)7-s + 1.04·8-s + (0.166 + 0.288i)9-s + (−0.331 − 0.191i)10-s + (1.28 + 0.744i)11-s + (−0.0549 + 0.0316i)12-s + 0.546·13-s + (−0.943 − 0.0150i)14-s + 0.234·15-s + (0.439 + 0.760i)16-s + (−0.187 + 0.982i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 357 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.264 - 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 357 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.264 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(357\)    =    \(3 \cdot 7 \cdot 17\)
Sign: $0.264 - 0.964i$
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: \((2,\ 357,\ (\ :1/2),\ 0.264 - 0.964i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.22442 + 0.933563i\)
\(L(\frac12)\) \(\approx\) \(1.22442 + 0.933563i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.35 - 2.26i)T \)
17 \( 1 + (0.772 - 4.05i)T \)
good2 \( 1 + (-0.667 - 1.15i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (0.786 - 0.453i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-4.27 - 2.46i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 1.97T + 13T^{2} \)
19 \( 1 + (0.0525 + 0.0910i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.89 + 2.82i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 5.49iT - 29T^{2} \)
31 \( 1 + (2.16 + 1.25i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.919 - 0.531i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 3.00iT - 41T^{2} \)
43 \( 1 + 9.82T + 43T^{2} \)
47 \( 1 + (-0.733 - 1.27i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.80 - 4.86i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.35 + 7.53i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.92 - 4.00i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.83 + 10.1i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 4.32iT - 71T^{2} \)
73 \( 1 + (0.228 + 0.131i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.84 - 1.06i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 13.0T + 83T^{2} \)
89 \( 1 + (4.37 + 7.57i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 4.99iT - 97T^{2} \)
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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

−11.69975621991319853589209617512, −10.91379789078063370884722629500, −9.799268677926750325651702554126, −8.736035210368450542384435073783, −7.52699111654780341337759495084, −6.53710898062128258412725278824, −6.14064847081250075677412202130, −4.95702867623429117452682776463, −3.77737377401232542940792805175, −1.77362480896687087936176321974, 1.13407131161117361597612846389, 3.27167994487892869635824206213, 3.91327300170946386970882096788, 4.98744723523522428540469909200, 6.50145719684151696318789133196, 7.26930707520717791529518418746, 8.624812321966620638991607164443, 9.678995127558155344636305447214, 10.70336966769949016504059141864, 11.39160451690550244460120165374

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