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Properties

Label 2-930-465.464-c1-0-46
Degree $2$
Conductor $930$
Sign $0.790 + 0.612i$
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  − 2-s + (1.70 − 0.288i)3-s + 4-s + (2.10 + 0.760i)5-s + (−1.70 + 0.288i)6-s − 4.63i·7-s − 8-s + (2.83 − 0.985i)9-s + (−2.10 − 0.760i)10-s + 1.87·11-s + (1.70 − 0.288i)12-s + 3.74·13-s + 4.63i·14-s + (3.81 + 0.692i)15-s + 16-s + 7.48i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.986 − 0.166i)3-s + 0.5·4-s + (0.940 + 0.340i)5-s + (−0.697 + 0.117i)6-s − 1.75i·7-s − 0.353·8-s + (0.944 − 0.328i)9-s + (−0.664 − 0.240i)10-s + 0.564·11-s + (0.493 − 0.0832i)12-s + 1.03·13-s + 1.23i·14-s + (0.983 + 0.178i)15-s + 0.250·16-s + 1.81i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.790 + 0.612i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.790 + 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.790 + 0.612i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (929, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ 0.790 + 0.612i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.90938 - 0.653028i\)
\(L(\frac12)\) \(\approx\) \(1.90938 - 0.653028i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + (-1.70 + 0.288i)T \)
5 \( 1 + (-2.10 - 0.760i)T \)
31 \( 1 + (4.94 + 2.56i)T \)
good7 \( 1 + 4.63iT - 7T^{2} \)
11 \( 1 - 1.87T + 11T^{2} \)
13 \( 1 - 3.74T + 13T^{2} \)
17 \( 1 - 7.48iT - 17T^{2} \)
19 \( 1 + 5.69T + 19T^{2} \)
23 \( 1 + 7.09iT - 23T^{2} \)
29 \( 1 + 6.19T + 29T^{2} \)
37 \( 1 - 1.89T + 37T^{2} \)
41 \( 1 - 3.77iT - 41T^{2} \)
43 \( 1 - 5.39T + 43T^{2} \)
47 \( 1 + 6.33T + 47T^{2} \)
53 \( 1 + 2.24iT - 53T^{2} \)
59 \( 1 - 4.58iT - 59T^{2} \)
61 \( 1 - 7.47iT - 61T^{2} \)
67 \( 1 - 8.18iT - 67T^{2} \)
71 \( 1 + 14.8iT - 71T^{2} \)
73 \( 1 - 11.5T + 73T^{2} \)
79 \( 1 - 10.6iT - 79T^{2} \)
83 \( 1 + 1.01iT - 83T^{2} \)
89 \( 1 - 11.8T + 89T^{2} \)
97 \( 1 + 1.87iT - 97T^{2} \)
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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

−10.01729617803890855282000215867, −9.080586923591443480948445164857, −8.394045748566933193881342905543, −7.58756900099793161326006140417, −6.58800767752823219488725750266, −6.23773263025718475627349386132, −4.19283954467515138327424203501, −3.62655682209280824378249725123, −2.10786013765548126032752083055, −1.23392016770991251852892984903, 1.66901469314976347585921108378, 2.38753421618964308441161497064, 3.48511832395501074680288894869, 5.06092014213929660523956151047, 5.88467035636443796482735047171, 6.82259198518621732630193382926, 7.970449174010550238037894211394, 8.844814970327878640030392638354, 9.264688837395930204605669646527, 9.558668845104199404274099179893

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