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Properties

Label 2-930-93.26-c1-0-3
Degree $2$
Conductor $930$
Sign $-0.0597 + 0.998i$
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 + (−1.06 + 1.36i)3-s − 4-s + (−0.866 + 0.5i)5-s + (−1.36 − 1.06i)6-s + (−2.35 + 4.08i)7-s − i·8-s + (−0.712 − 2.91i)9-s + (−0.5 − 0.866i)10-s + (3.18 + 5.51i)11-s + (1.06 − 1.36i)12-s + (−2.96 + 1.71i)13-s + (−4.08 − 2.35i)14-s + (0.244 − 1.71i)15-s + 16-s + (1.65 − 2.87i)17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.617 + 0.786i)3-s − 0.5·4-s + (−0.387 + 0.223i)5-s + (−0.556 − 0.436i)6-s + (−0.890 + 1.54i)7-s − 0.353i·8-s + (−0.237 − 0.971i)9-s + (−0.158 − 0.273i)10-s + (0.960 + 1.66i)11-s + (0.308 − 0.393i)12-s + (−0.823 + 0.475i)13-s + (−1.09 − 0.629i)14-s + (0.0632 − 0.442i)15-s + 0.250·16-s + (0.402 − 0.696i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0597 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0597 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.0597 + 0.998i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (491, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ -0.0597 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.364657 - 0.387145i\)
\(L(\frac12)\) \(\approx\) \(0.364657 - 0.387145i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.06 - 1.36i)T \)
5 \( 1 + (0.866 - 0.5i)T \)
31 \( 1 + (-3.67 + 4.18i)T \)
good7 \( 1 + (2.35 - 4.08i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-3.18 - 5.51i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.96 - 1.71i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.65 + 2.87i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.56 - 4.43i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 4.89T + 23T^{2} \)
29 \( 1 - 5.57T + 29T^{2} \)
37 \( 1 + (4.26 + 2.46i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-8.33 + 4.81i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.26 + 3.04i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 3.50iT - 47T^{2} \)
53 \( 1 + (-0.645 - 1.11i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.87 + 2.81i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 - 1.31iT - 61T^{2} \)
67 \( 1 + (-7.24 - 12.5i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (1.61 - 0.929i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-9.05 + 5.22i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.954 - 0.551i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.19 - 5.54i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 6.30T + 89T^{2} \)
97 \( 1 - 0.469T + 97T^{2} \)
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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

−10.34541510429782035190751375609, −9.560649292840670961174692305470, −9.367791432477013756505797426608, −8.221487925046884698205629990395, −7.00865045312737212771837517789, −6.39379709365752429766268077313, −5.57859179219449201093541598117, −4.60667030699198792908188160897, −3.81669469538144289138101030240, −2.37639082260930872714973058935, 0.31604468067709433281228404364, 1.14279911949941980352714683608, 2.95865689659082725951446194137, 3.86325180793707130732494685001, 4.86860941751644260637263671248, 6.21858774229047607099018521584, 6.71207393758039984438887709148, 7.84569003278066564510637595444, 8.487375129832272312319616547021, 9.701691356942448365340252799366

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