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Properties

Label 2-930-15.8-c1-0-42
Degree $2$
Conductor $930$
Sign $-0.208 + 0.977i$
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 + (−1.46 − 0.919i)3-s + 1.00i·4-s + (−2.22 + 0.261i)5-s + (−0.387 − 1.68i)6-s + (3.32 − 3.32i)7-s + (−0.707 + 0.707i)8-s + (1.30 + 2.69i)9-s + (−1.75 − 1.38i)10-s + 3.28i·11-s + (0.919 − 1.46i)12-s + (−1.75 − 1.75i)13-s + 4.70·14-s + (3.50 + 1.65i)15-s − 1.00·16-s + (−5.45 − 5.45i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (−0.847 − 0.530i)3-s + 0.500i·4-s + (−0.993 + 0.117i)5-s + (−0.158 − 0.689i)6-s + (1.25 − 1.25i)7-s + (−0.250 + 0.250i)8-s + (0.436 + 0.899i)9-s + (−0.555 − 0.438i)10-s + 0.989i·11-s + (0.265 − 0.423i)12-s + (−0.486 − 0.486i)13-s + 1.25·14-s + (0.903 + 0.427i)15-s − 0.250·16-s + (−1.32 − 1.32i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.208 + 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.208 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.208 + 0.977i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (683, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ -0.208 + 0.977i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.504590 - 0.623713i\)
\(L(\frac12)\) \(\approx\) \(0.504590 - 0.623713i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.46 + 0.919i)T \)
5 \( 1 + (2.22 - 0.261i)T \)
31 \( 1 - T \)
good7 \( 1 + (-3.32 + 3.32i)T - 7iT^{2} \)
11 \( 1 - 3.28iT - 11T^{2} \)
13 \( 1 + (1.75 + 1.75i)T + 13iT^{2} \)
17 \( 1 + (5.45 + 5.45i)T + 17iT^{2} \)
19 \( 1 + 0.692iT - 19T^{2} \)
23 \( 1 + (1.40 - 1.40i)T - 23iT^{2} \)
29 \( 1 + 1.76T + 29T^{2} \)
37 \( 1 + (-2.58 + 2.58i)T - 37iT^{2} \)
41 \( 1 + 6.83iT - 41T^{2} \)
43 \( 1 + (8.40 + 8.40i)T + 43iT^{2} \)
47 \( 1 + (8.79 + 8.79i)T + 47iT^{2} \)
53 \( 1 + (-4.69 + 4.69i)T - 53iT^{2} \)
59 \( 1 - 6.39T + 59T^{2} \)
61 \( 1 + 12.2T + 61T^{2} \)
67 \( 1 + (7.69 - 7.69i)T - 67iT^{2} \)
71 \( 1 - 0.464iT - 71T^{2} \)
73 \( 1 + (-4.31 - 4.31i)T + 73iT^{2} \)
79 \( 1 + 14.1iT - 79T^{2} \)
83 \( 1 + (-0.133 + 0.133i)T - 83iT^{2} \)
89 \( 1 - 5.90T + 89T^{2} \)
97 \( 1 + (2.86 - 2.86i)T - 97iT^{2} \)
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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

−10.14742450506802487650352945003, −8.624451078456934704844705488727, −7.61816688492129548066901915393, −7.29698922521243409154905237812, −6.73845602982702151072748972877, −5.09897130714842263969396464032, −4.76694338440223802620899125466, −3.88699627543562368914582574924, −2.09796644296645228138627798471, −0.35848938325112881150846353951, 1.60348412669745491085281525942, 3.09322697174122962340661974929, 4.40179077975068867776045959290, 4.71198949517372214660132699362, 5.83215168053704284295382058686, 6.50506208816564430670419114939, 8.042532397452030607682403225379, 8.626073635292591661623186541290, 9.536580857210484904441552010536, 10.83085612360017256132069350427

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