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Properties

Label 2-930-15.2-c1-0-34
Degree $2$
Conductor $930$
Sign $0.989 + 0.142i$
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  + (0.707 − 0.707i)2-s + (1.73 − 0.0444i)3-s − 1.00i·4-s + (0.142 + 2.23i)5-s + (1.19 − 1.25i)6-s + (2.34 + 2.34i)7-s + (−0.707 − 0.707i)8-s + (2.99 − 0.153i)9-s + (1.67 + 1.47i)10-s − 5.32i·11-s + (−0.0444 − 1.73i)12-s + (−2.57 + 2.57i)13-s + 3.31·14-s + (0.345 + 3.85i)15-s − 1.00·16-s + (2.33 − 2.33i)17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (0.999 − 0.0256i)3-s − 0.500i·4-s + (0.0635 + 0.997i)5-s + (0.487 − 0.512i)6-s + (0.885 + 0.885i)7-s + (−0.250 − 0.250i)8-s + (0.998 − 0.0512i)9-s + (0.530 + 0.467i)10-s − 1.60i·11-s + (−0.0128 − 0.499i)12-s + (−0.714 + 0.714i)13-s + 0.885·14-s + (0.0891 + 0.996i)15-s − 0.250·16-s + (0.565 − 0.565i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.142i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 + 0.142i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.989 + 0.142i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (497, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ 0.989 + 0.142i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.15298 - 0.225091i\)
\(L(\frac12)\) \(\approx\) \(3.15298 - 0.225091i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 0.707i)T \)
3 \( 1 + (-1.73 + 0.0444i)T \)
5 \( 1 + (-0.142 - 2.23i)T \)
31 \( 1 - T \)
good7 \( 1 + (-2.34 - 2.34i)T + 7iT^{2} \)
11 \( 1 + 5.32iT - 11T^{2} \)
13 \( 1 + (2.57 - 2.57i)T - 13iT^{2} \)
17 \( 1 + (-2.33 + 2.33i)T - 17iT^{2} \)
19 \( 1 - 5.87iT - 19T^{2} \)
23 \( 1 + (2.05 + 2.05i)T + 23iT^{2} \)
29 \( 1 - 4.07T + 29T^{2} \)
37 \( 1 + (-0.718 - 0.718i)T + 37iT^{2} \)
41 \( 1 - 9.62iT - 41T^{2} \)
43 \( 1 + (-1.71 + 1.71i)T - 43iT^{2} \)
47 \( 1 + (-6.49 + 6.49i)T - 47iT^{2} \)
53 \( 1 + (5.55 + 5.55i)T + 53iT^{2} \)
59 \( 1 + 6.86T + 59T^{2} \)
61 \( 1 + 0.913T + 61T^{2} \)
67 \( 1 + (9.71 + 9.71i)T + 67iT^{2} \)
71 \( 1 + 8.59iT - 71T^{2} \)
73 \( 1 + (5.80 - 5.80i)T - 73iT^{2} \)
79 \( 1 + 10.0iT - 79T^{2} \)
83 \( 1 + (4.35 + 4.35i)T + 83iT^{2} \)
89 \( 1 + 3.15T + 89T^{2} \)
97 \( 1 + (7.10 + 7.10i)T + 97iT^{2} \)
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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

−10.11162995318009735750150970004, −9.278716465860502668132768575488, −8.340959877875778179004961528911, −7.73261721125192979884911165620, −6.51209782986654298606610009294, −5.68999715615854525271179344214, −4.52533570251799899678591048455, −3.38362560541089456651844354437, −2.70553060101944157748385982320, −1.71014034433388636229320403678, 1.40900534640446866025593633050, 2.65545659559590415387867340472, 4.26376530240167239541139640026, 4.48261353878351139699736022564, 5.46861407324701283313509455132, 7.06351129759900836476128753285, 7.57742988435124230564812084160, 8.176345184338689202656306292478, 9.154548569257200190179615306213, 9.889373118396265317558705023817

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