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Properties

Label 2-930-155.39-c1-0-26
Degree $2$
Conductor $930$
Sign $0.844 + 0.535i$
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 + (−2.22 + 0.254i)5-s + 0.999·6-s + (2.23 − 3.07i)7-s + (0.587 + 0.809i)8-s + (0.809 − 0.587i)9-s + (−2.19 − 0.444i)10-s + (−2.25 − 1.64i)11-s + (0.951 + 0.309i)12-s + (3.92 − 1.27i)13-s + (3.07 − 2.23i)14-s + (−2.03 + 0.928i)15-s + (0.309 + 0.951i)16-s + (0.619 + 0.853i)17-s + ⋯
L(s)  = 1  + (0.672 + 0.218i)2-s + (0.549 − 0.178i)3-s + (0.404 + 0.293i)4-s + (−0.993 + 0.113i)5-s + 0.408·6-s + (0.844 − 1.16i)7-s + (0.207 + 0.286i)8-s + (0.269 − 0.195i)9-s + (−0.693 − 0.140i)10-s + (−0.681 − 0.494i)11-s + (0.274 + 0.0892i)12-s + (1.08 − 0.353i)13-s + (0.821 − 0.596i)14-s + (−0.525 + 0.239i)15-s + (0.0772 + 0.237i)16-s + (0.150 + 0.206i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.844 + 0.535i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.844 + 0.535i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.844 + 0.535i$
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: \((2,\ 930,\ (\ :1/2),\ 0.844 + 0.535i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.52861 - 0.734805i\)
\(L(\frac12)\) \(\approx\) \(2.52861 - 0.734805i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.22 - 0.254i)T \)
31 \( 1 + (-2.85 - 4.77i)T \)
good7 \( 1 + (-2.23 + 3.07i)T + (-2.16 - 6.65i)T^{2} \)
11 \( 1 + (2.25 + 1.64i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (-3.92 + 1.27i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (-0.619 - 0.853i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (-0.925 + 2.84i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (0.909 + 1.25i)T + (-7.10 + 21.8i)T^{2} \)
29 \( 1 + (-0.0548 + 0.168i)T + (-23.4 - 17.0i)T^{2} \)
37 \( 1 + 2.89iT - 37T^{2} \)
41 \( 1 + (-2.53 + 7.80i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + (-1.12 - 0.364i)T + (34.7 + 25.2i)T^{2} \)
47 \( 1 + (8.77 - 2.85i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (-4.06 - 5.59i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (-3.06 - 9.42i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 - 2.03T + 61T^{2} \)
67 \( 1 - 9.47iT - 67T^{2} \)
71 \( 1 + (-10.1 + 7.35i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (3.27 - 4.51i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (10.2 - 7.43i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (2.80 + 0.912i)T + (67.1 + 48.7i)T^{2} \)
89 \( 1 + (7.02 + 5.10i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (3.10 - 4.26i)T + (-29.9 - 92.2i)T^{2} \)
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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

−10.34402610040159531210417880312, −8.713079098181962136904159689659, −8.160202126450181016587847145001, −7.48415927197613473655139327132, −6.79392369873485345187383434314, −5.52303432157331703630621412552, −4.42680481760761109694088417121, −3.77581391766161953455551968379, −2.82448034354655618912975906724, −1.05724092941956088068120439918, 1.68407998810796379565625953099, 2.83676799510007019899446293601, 3.84611339387777775743524538654, 4.74329062681986141091918009277, 5.52760360581544704386208073875, 6.68026897941529091717346399793, 8.025130670926855874716669121990, 8.138353747953421913189681082697, 9.260167021840216141728467073068, 10.20983092148499104105239115140

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