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Properties

Label 2-930-155.39-c1-0-15
Degree $2$
Conductor $930$
Sign $0.999 - 0.0191i$
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  + (−0.951 − 0.309i)2-s + (0.951 − 0.309i)3-s + (0.809 + 0.587i)4-s + (2.21 + 0.332i)5-s − 0.999·6-s + (−0.907 + 1.24i)7-s + (−0.587 − 0.809i)8-s + (0.809 − 0.587i)9-s + (−2.00 − 0.999i)10-s + (1.74 + 1.27i)11-s + (0.951 + 0.309i)12-s + (0.363 − 0.118i)13-s + (1.24 − 0.907i)14-s + (2.20 − 0.367i)15-s + (0.309 + 0.951i)16-s + (−0.638 − 0.879i)17-s + ⋯
L(s)  = 1  + (−0.672 − 0.218i)2-s + (0.549 − 0.178i)3-s + (0.404 + 0.293i)4-s + (0.988 + 0.148i)5-s − 0.408·6-s + (−0.342 + 0.471i)7-s + (−0.207 − 0.286i)8-s + (0.269 − 0.195i)9-s + (−0.632 − 0.315i)10-s + (0.527 + 0.383i)11-s + (0.274 + 0.0892i)12-s + (0.100 − 0.0327i)13-s + (0.333 − 0.242i)14-s + (0.569 − 0.0948i)15-s + (0.0772 + 0.237i)16-s + (−0.154 − 0.213i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0191i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0191i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.999 - 0.0191i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ 0.999 - 0.0191i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.69927 + 0.0162958i\)
\(L(\frac12)\) \(\approx\) \(1.69927 + 0.0162958i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.951 + 0.309i)T \)
3 \( 1 + (-0.951 + 0.309i)T \)
5 \( 1 + (-2.21 - 0.332i)T \)
31 \( 1 + (-5.47 - 1.01i)T \)
good7 \( 1 + (0.907 - 1.24i)T + (-2.16 - 6.65i)T^{2} \)
11 \( 1 + (-1.74 - 1.27i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (-0.363 + 0.118i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (0.638 + 0.879i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (-1.32 + 4.07i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (-2.62 - 3.61i)T + (-7.10 + 21.8i)T^{2} \)
29 \( 1 + (1.31 - 4.05i)T + (-23.4 - 17.0i)T^{2} \)
37 \( 1 + 3.70iT - 37T^{2} \)
41 \( 1 + (-0.587 + 1.80i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + (-7.37 - 2.39i)T + (34.7 + 25.2i)T^{2} \)
47 \( 1 + (9.01 - 2.92i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (3.47 + 4.78i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (-1.08 - 3.33i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + 4.97T + 61T^{2} \)
67 \( 1 - 2.20iT - 67T^{2} \)
71 \( 1 + (4.31 - 3.13i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-3.86 + 5.32i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (-4.01 + 2.91i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-1.57 - 0.512i)T + (67.1 + 48.7i)T^{2} \)
89 \( 1 + (0.341 + 0.248i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-8.18 + 11.2i)T + (-29.9 - 92.2i)T^{2} \)
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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

−9.761442010979827369227799451311, −9.260454904204058646978585308247, −8.759376609800249569739815348067, −7.53829993050422790328049880246, −6.80314647584084296106933038687, −5.98193237504908822618249028073, −4.79387960245374624862144133114, −3.27000575376514930390129295669, −2.45492731578053504680816572044, −1.33160153826247365721837310942, 1.13846250617316543489850696159, 2.38463666876167301779126593242, 3.57670165840534032856495567183, 4.81435420728876564977723591619, 6.06065416903890276257594000583, 6.58608800024629245374314041128, 7.70765429066027540574833054265, 8.538328017705094465383154903452, 9.246634211328836763550963578743, 9.968696241467009386206276869321

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