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Properties

Label 2-930-15.2-c1-0-47
Degree $2$
Conductor $930$
Sign $-0.586 + 0.810i$
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 + (−0.292 − 1.70i)3-s − 1.00i·4-s + (1.19 − 1.88i)5-s + (−1.41 − 0.999i)6-s + (3.59 + 3.59i)7-s + (−0.707 − 0.707i)8-s + (−2.82 + i)9-s + (−0.489 − 2.18i)10-s − 1.67i·11-s + (−1.70 + 0.292i)12-s + (−0.721 + 0.721i)13-s + 5.08·14-s + (−3.57 − 1.48i)15-s − 1.00·16-s + (5.06 − 5.06i)17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (−0.169 − 0.985i)3-s − 0.500i·4-s + (0.535 − 0.844i)5-s + (−0.577 − 0.408i)6-s + (1.35 + 1.35i)7-s + (−0.250 − 0.250i)8-s + (−0.942 + 0.333i)9-s + (−0.154 − 0.689i)10-s − 0.503i·11-s + (−0.492 + 0.0845i)12-s + (−0.200 + 0.200i)13-s + 1.35·14-s + (−0.923 − 0.384i)15-s − 0.250·16-s + (1.22 − 1.22i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.586 + 0.810i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.586 + 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.586 + 0.810i$
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: \((2,\ 930,\ (\ :1/2),\ -0.586 + 0.810i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.06440 - 2.08457i\)
\(L(\frac12)\) \(\approx\) \(1.06440 - 2.08457i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.292 + 1.70i)T \)
5 \( 1 + (-1.19 + 1.88i)T \)
31 \( 1 - T \)
good7 \( 1 + (-3.59 - 3.59i)T + 7iT^{2} \)
11 \( 1 + 1.67iT - 11T^{2} \)
13 \( 1 + (0.721 - 0.721i)T - 13iT^{2} \)
17 \( 1 + (-5.06 + 5.06i)T - 17iT^{2} \)
19 \( 1 + 7.32iT - 19T^{2} \)
23 \( 1 + (-3.01 - 3.01i)T + 23iT^{2} \)
29 \( 1 + 10.0T + 29T^{2} \)
37 \( 1 + (2 + 2i)T + 37iT^{2} \)
41 \( 1 - 10.6iT - 41T^{2} \)
43 \( 1 + (5.28 - 5.28i)T - 43iT^{2} \)
47 \( 1 + (-1.86 + 1.86i)T - 47iT^{2} \)
53 \( 1 + (-0.190 - 0.190i)T + 53iT^{2} \)
59 \( 1 - 3.00T + 59T^{2} \)
61 \( 1 - 9.83T + 61T^{2} \)
67 \( 1 + (-1.37 - 1.37i)T + 67iT^{2} \)
71 \( 1 + 1.32iT - 71T^{2} \)
73 \( 1 + (0.424 - 0.424i)T - 73iT^{2} \)
79 \( 1 + 1.39iT - 79T^{2} \)
83 \( 1 + (7.37 + 7.37i)T + 83iT^{2} \)
89 \( 1 - 3.30T + 89T^{2} \)
97 \( 1 + (-5.96 - 5.96i)T + 97iT^{2} \)
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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

−9.537349474000211081240006571133, −8.993144664826511112978583157419, −8.173219801353317643211096253116, −7.25393999690011890769544100198, −5.97208041883878869290080850999, −5.22752373424139684707628444503, −4.93957545262454561333990068740, −2.93559111293041560026696835628, −2.05381755544407344082691857951, −1.05066837247560672291270551446, 1.82362478228842445402596689552, 3.59046002488113207137785174873, 4.00006799398940030147514680323, 5.22241067153090614936201684651, 5.76354407255397695023535392772, 6.98931918002509199458238930083, 7.72695581170317134801678885581, 8.489216102708037422064062493658, 9.895060781549577074632498024640, 10.37471570741925347847405742220

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