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Properties

Label 2-930-15.8-c1-0-45
Degree $2$
Conductor $930$
Sign $-0.945 - 0.325i$
Analytic cond. $7.42608$
Root an. cond. $2.72508$
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  + (−0.707 − 0.707i)2-s + (−1.56 − 0.750i)3-s + 1.00i·4-s + (0.256 − 2.22i)5-s + (0.573 + 1.63i)6-s + (0.463 − 0.463i)7-s + (0.707 − 0.707i)8-s + (1.87 + 2.34i)9-s + (−1.75 + 1.38i)10-s − 0.686i·11-s + (0.750 − 1.56i)12-s + (−0.948 − 0.948i)13-s − 0.655·14-s + (−2.06 + 3.27i)15-s − 1.00·16-s + (−3.31 − 3.31i)17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (−0.901 − 0.433i)3-s + 0.500i·4-s + (0.114 − 0.993i)5-s + (0.233 + 0.667i)6-s + (0.175 − 0.175i)7-s + (0.250 − 0.250i)8-s + (0.624 + 0.781i)9-s + (−0.554 + 0.439i)10-s − 0.206i·11-s + (0.216 − 0.450i)12-s + (−0.263 − 0.263i)13-s − 0.175·14-s + (−0.533 + 0.845i)15-s − 0.250·16-s + (−0.803 − 0.803i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.945 - 0.325i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.945 - 0.325i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.945 - 0.325i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (683, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ -0.945 - 0.325i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0897337 + 0.536575i\)
\(L(\frac12)\) \(\approx\) \(0.0897337 + 0.536575i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 + 0.707i)T \)
3 \( 1 + (1.56 + 0.750i)T \)
5 \( 1 + (-0.256 + 2.22i)T \)
31 \( 1 + T \)
good7 \( 1 + (-0.463 + 0.463i)T - 7iT^{2} \)
11 \( 1 + 0.686iT - 11T^{2} \)
13 \( 1 + (0.948 + 0.948i)T + 13iT^{2} \)
17 \( 1 + (3.31 + 3.31i)T + 17iT^{2} \)
19 \( 1 + 0.717iT - 19T^{2} \)
23 \( 1 + (-6.73 + 6.73i)T - 23iT^{2} \)
29 \( 1 - 5.30T + 29T^{2} \)
37 \( 1 + (5.56 - 5.56i)T - 37iT^{2} \)
41 \( 1 + 2.24iT - 41T^{2} \)
43 \( 1 + (-4.06 - 4.06i)T + 43iT^{2} \)
47 \( 1 + (8.61 + 8.61i)T + 47iT^{2} \)
53 \( 1 + (5.32 - 5.32i)T - 53iT^{2} \)
59 \( 1 + 13.6T + 59T^{2} \)
61 \( 1 + 13.2T + 61T^{2} \)
67 \( 1 + (-3.26 + 3.26i)T - 67iT^{2} \)
71 \( 1 - 6.49iT - 71T^{2} \)
73 \( 1 + (-1.62 - 1.62i)T + 73iT^{2} \)
79 \( 1 + 5.04iT - 79T^{2} \)
83 \( 1 + (-6.41 + 6.41i)T - 83iT^{2} \)
89 \( 1 - 4.84T + 89T^{2} \)
97 \( 1 + (11.0 - 11.0i)T - 97iT^{2} \)
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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

−9.625162428325988225300812578650, −8.820258548290310420637546556596, −8.045043157209139453334013964793, −7.07914576717258707104386439373, −6.24835894333136298601881838209, −4.91629635987990177258130032315, −4.60445414664596596602629478481, −2.80546825708447317442812254563, −1.44505644568430481692740689893, −0.35751712257495466719963273658, 1.70608239050808326680407559534, 3.31959890536277206228600596971, 4.56374651168867043244207930941, 5.49272588846085879242351308006, 6.39316988933379977035394109543, 6.95555440919138951929319139246, 7.84080924604428908824949939006, 9.101034491495874604644423041900, 9.662603366988593821959386094086, 10.68289023914954201361934805329

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