Properties

Label 2-930-31.19-c1-0-6
Degree $2$
Conductor $930$
Sign $0.524 - 0.851i$
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  + (−0.809 − 0.587i)2-s + (0.913 + 0.406i)3-s + (0.309 + 0.951i)4-s + (−0.5 + 0.866i)5-s + (−0.5 − 0.866i)6-s + (0.955 − 1.06i)7-s + (0.309 − 0.951i)8-s + (0.669 + 0.743i)9-s + (0.913 − 0.406i)10-s + (−4.33 − 0.920i)11-s + (−0.104 + 0.994i)12-s + (0.559 + 5.32i)13-s + (−1.39 + 0.297i)14-s + (−0.809 + 0.587i)15-s + (−0.809 + 0.587i)16-s + (5.84 − 1.24i)17-s + ⋯
L(s)  = 1  + (−0.572 − 0.415i)2-s + (0.527 + 0.234i)3-s + (0.154 + 0.475i)4-s + (−0.223 + 0.387i)5-s + (−0.204 − 0.353i)6-s + (0.361 − 0.401i)7-s + (0.109 − 0.336i)8-s + (0.223 + 0.247i)9-s + (0.288 − 0.128i)10-s + (−1.30 − 0.277i)11-s + (−0.0301 + 0.287i)12-s + (0.155 + 1.47i)13-s + (−0.373 + 0.0793i)14-s + (−0.208 + 0.151i)15-s + (−0.202 + 0.146i)16-s + (1.41 − 0.301i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.524 - 0.851i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.524 - 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.524 - 0.851i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (391, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ 0.524 - 0.851i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.06997 + 0.597973i\)
\(L(\frac12)\) \(\approx\) \(1.06997 + 0.597973i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.809 + 0.587i)T \)
3 \( 1 + (-0.913 - 0.406i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
31 \( 1 + (-5.13 - 2.14i)T \)
good7 \( 1 + (-0.955 + 1.06i)T + (-0.731 - 6.96i)T^{2} \)
11 \( 1 + (4.33 + 0.920i)T + (10.0 + 4.47i)T^{2} \)
13 \( 1 + (-0.559 - 5.32i)T + (-12.7 + 2.70i)T^{2} \)
17 \( 1 + (-5.84 + 1.24i)T + (15.5 - 6.91i)T^{2} \)
19 \( 1 + (0.590 - 5.61i)T + (-18.5 - 3.95i)T^{2} \)
23 \( 1 + (0.413 - 1.27i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (-0.774 - 0.562i)T + (8.96 + 27.5i)T^{2} \)
37 \( 1 + (3.16 + 5.48i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.41 + 1.07i)T + (27.4 - 30.4i)T^{2} \)
43 \( 1 + (0.908 - 8.64i)T + (-42.0 - 8.94i)T^{2} \)
47 \( 1 + (9.73 - 7.07i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-1.81 - 2.01i)T + (-5.54 + 52.7i)T^{2} \)
59 \( 1 + (-2.83 - 1.26i)T + (39.4 + 43.8i)T^{2} \)
61 \( 1 - 8.56T + 61T^{2} \)
67 \( 1 + (6.36 - 11.0i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.74 + 3.05i)T + (-7.42 + 70.6i)T^{2} \)
73 \( 1 + (-3.77 - 0.802i)T + (66.6 + 29.6i)T^{2} \)
79 \( 1 + (-12.9 + 2.76i)T + (72.1 - 32.1i)T^{2} \)
83 \( 1 + (4.81 - 2.14i)T + (55.5 - 61.6i)T^{2} \)
89 \( 1 + (2.77 + 8.53i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (2.63 + 8.12i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.14030434014492392434293189566, −9.553664262712696104892654799125, −8.431409700543896744547779301033, −7.86221399953206646650638739209, −7.19379698616192979643914450121, −5.94405460122060412828637233255, −4.67223919531086976167656924807, −3.67035794898093721820774967950, −2.75161887219035385344592072352, −1.48442150696972292746450852978, 0.68727882237931437062567542501, 2.28276566316649468050622378664, 3.31492013951612604630174901559, 4.97705569115661804201861366421, 5.46548759255565622749004409234, 6.71894215144659544422351625764, 7.84047849398792119058542050674, 8.081860031209679123545513760555, 8.817429618582982002386352390795, 10.01418362658305264381219584944

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