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Properties

Label 2-930-93.92-c1-0-4
Degree $2$
Conductor $930$
Sign $-0.990 + 0.138i$
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  + i·2-s + (−1.71 + 0.206i)3-s − 4-s − i·5-s + (−0.206 − 1.71i)6-s − 1.18·7-s − i·8-s + (2.91 − 0.711i)9-s + 10-s + 1.57·11-s + (1.71 − 0.206i)12-s + 3.43i·13-s − 1.18i·14-s + (0.206 + 1.71i)15-s + 16-s − 4.55·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.992 + 0.119i)3-s − 0.5·4-s − 0.447i·5-s + (−0.0844 − 0.702i)6-s − 0.446·7-s − 0.353i·8-s + (0.971 − 0.237i)9-s + 0.316·10-s + 0.475·11-s + (0.496 − 0.0596i)12-s + 0.953i·13-s − 0.315i·14-s + (0.0533 + 0.444i)15-s + 0.250·16-s − 1.10·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.138i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 + 0.138i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.990 + 0.138i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (371, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ -0.990 + 0.138i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0235477 - 0.339207i\)
\(L(\frac12)\) \(\approx\) \(0.0235477 - 0.339207i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 + (1.71 - 0.206i)T \)
5 \( 1 + iT \)
31 \( 1 + (5.56 - 0.105i)T \)
good7 \( 1 + 1.18T + 7T^{2} \)
11 \( 1 - 1.57T + 11T^{2} \)
13 \( 1 - 3.43iT - 13T^{2} \)
17 \( 1 + 4.55T + 17T^{2} \)
19 \( 1 - 6.64T + 19T^{2} \)
23 \( 1 + 1.16T + 23T^{2} \)
29 \( 1 + 8.20T + 29T^{2} \)
37 \( 1 - 9.82iT - 37T^{2} \)
41 \( 1 - 5.42iT - 41T^{2} \)
43 \( 1 + 6.82iT - 43T^{2} \)
47 \( 1 + 1.46iT - 47T^{2} \)
53 \( 1 + 7.16T + 53T^{2} \)
59 \( 1 + 6.84iT - 59T^{2} \)
61 \( 1 - 11.8iT - 61T^{2} \)
67 \( 1 + 13.6T + 67T^{2} \)
71 \( 1 - 6.52iT - 71T^{2} \)
73 \( 1 + 4.67iT - 73T^{2} \)
79 \( 1 + 6.13iT - 79T^{2} \)
83 \( 1 + 3.86T + 83T^{2} \)
89 \( 1 - 10.2T + 89T^{2} \)
97 \( 1 - 3.46T + 97T^{2} \)
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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

−10.40940067881795330598862981186, −9.374500848929041985567587331030, −9.147768566373307418965372017936, −7.74914146849527296681438656579, −6.91128839711751255695112927772, −6.27703924186082631492252412130, −5.35796136666482976955151484949, −4.55720172351874911348039990740, −3.62768266241237767251332948933, −1.52037332471075790043936588276, 0.18840570678276865232623881201, 1.75072472080850813035683776648, 3.17830732435778352444674438839, 4.12602044765358767106425118670, 5.32605575178602582408018181324, 5.99692469555715612011149560474, 7.07091480731141522189951252119, 7.76897943736082574475490366944, 9.248387995839264010951345860319, 9.690002280245025606241631713377

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