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Properties

Label 2-930-15.2-c1-0-44
Degree $2$
Conductor $930$
Sign $-0.788 + 0.614i$
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 + (−1.50 − 0.853i)3-s − 1.00i·4-s + (2.20 + 0.358i)5-s + (−1.66 + 0.461i)6-s + (−1.72 − 1.72i)7-s + (−0.707 − 0.707i)8-s + (1.54 + 2.57i)9-s + (1.81 − 1.30i)10-s + 0.339i·11-s + (−0.853 + 1.50i)12-s + (2.77 − 2.77i)13-s − 2.43·14-s + (−3.01 − 2.42i)15-s − 1.00·16-s + (0.287 − 0.287i)17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (−0.870 − 0.492i)3-s − 0.500i·4-s + (0.987 + 0.160i)5-s + (−0.681 + 0.188i)6-s + (−0.650 − 0.650i)7-s + (−0.250 − 0.250i)8-s + (0.513 + 0.857i)9-s + (0.573 − 0.413i)10-s + 0.102i·11-s + (−0.246 + 0.435i)12-s + (0.769 − 0.769i)13-s − 0.650·14-s + (−0.779 − 0.626i)15-s − 0.250·16-s + (0.0697 − 0.0697i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.788 + 0.614i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.788 + 0.614i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.788 + 0.614i$
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: \((2,\ 930,\ (\ :1/2),\ -0.788 + 0.614i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.487555 - 1.41817i\)
\(L(\frac12)\) \(\approx\) \(0.487555 - 1.41817i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.50 + 0.853i)T \)
5 \( 1 + (-2.20 - 0.358i)T \)
31 \( 1 - T \)
good7 \( 1 + (1.72 + 1.72i)T + 7iT^{2} \)
11 \( 1 - 0.339iT - 11T^{2} \)
13 \( 1 + (-2.77 + 2.77i)T - 13iT^{2} \)
17 \( 1 + (-0.287 + 0.287i)T - 17iT^{2} \)
19 \( 1 + 5.47iT - 19T^{2} \)
23 \( 1 + (3.99 + 3.99i)T + 23iT^{2} \)
29 \( 1 + 6.27T + 29T^{2} \)
37 \( 1 + (-0.324 - 0.324i)T + 37iT^{2} \)
41 \( 1 - 8.78iT - 41T^{2} \)
43 \( 1 + (-6.31 + 6.31i)T - 43iT^{2} \)
47 \( 1 + (2.27 - 2.27i)T - 47iT^{2} \)
53 \( 1 + (7.53 + 7.53i)T + 53iT^{2} \)
59 \( 1 + 0.628T + 59T^{2} \)
61 \( 1 + 11.4T + 61T^{2} \)
67 \( 1 + (-0.339 - 0.339i)T + 67iT^{2} \)
71 \( 1 + 1.05iT - 71T^{2} \)
73 \( 1 + (-6.50 + 6.50i)T - 73iT^{2} \)
79 \( 1 - 3.87iT - 79T^{2} \)
83 \( 1 + (-6.20 - 6.20i)T + 83iT^{2} \)
89 \( 1 - 13.1T + 89T^{2} \)
97 \( 1 + (-0.395 - 0.395i)T + 97iT^{2} \)
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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

−10.02661243915585102057280262518, −9.235418135867579038445787707144, −7.87990858121196258271190787992, −6.76902443579806131595460842393, −6.25662618578427231389721325902, −5.44527949704289273868167051072, −4.49778474348971962426143738475, −3.19957146485833279658074732261, −2.00269642823697272005884001026, −0.66887923142501563366739928959, 1.75501837017810069746351374669, 3.37989782115989442656431049343, 4.30564284402637870725763270823, 5.52213464752594687280957192042, 5.95644337081928787989291637608, 6.48427510562740520199995418106, 7.71698070568784363897213871479, 9.055208559586480861568355235117, 9.428310814983316796037638472483, 10.36620487840685122672122832632

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