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Properties

Label 2-930-31.2-c1-0-13
Degree $2$
Conductor $930$
Sign $0.678 + 0.734i$
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 + (−0.809 + 0.587i)3-s + (0.309 + 0.951i)4-s + 5-s + 6-s + (−0.530 − 1.63i)7-s + (0.309 − 0.951i)8-s + (0.309 − 0.951i)9-s + (−0.809 − 0.587i)10-s + (0.890 + 2.74i)11-s + (−0.809 − 0.587i)12-s + (0.180 − 0.131i)13-s + (−0.530 + 1.63i)14-s + (−0.809 + 0.587i)15-s + (−0.809 + 0.587i)16-s + (1.60 − 4.93i)17-s + ⋯
L(s)  = 1  + (−0.572 − 0.415i)2-s + (−0.467 + 0.339i)3-s + (0.154 + 0.475i)4-s + 0.447·5-s + 0.408·6-s + (−0.200 − 0.617i)7-s + (0.109 − 0.336i)8-s + (0.103 − 0.317i)9-s + (−0.255 − 0.185i)10-s + (0.268 + 0.826i)11-s + (−0.233 − 0.169i)12-s + (0.0501 − 0.0364i)13-s + (−0.141 + 0.436i)14-s + (−0.208 + 0.151i)15-s + (−0.202 + 0.146i)16-s + (0.388 − 1.19i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.678 + 0.734i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.678 + 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.678 + 0.734i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (901, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ 0.678 + 0.734i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.953817 - 0.417676i\)
\(L(\frac12)\) \(\approx\) \(0.953817 - 0.417676i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.809 - 0.587i)T \)
5 \( 1 - T \)
31 \( 1 + (-0.731 - 5.51i)T \)
good7 \( 1 + (0.530 + 1.63i)T + (-5.66 + 4.11i)T^{2} \)
11 \( 1 + (-0.890 - 2.74i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (-0.180 + 0.131i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-1.60 + 4.93i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-0.261 - 0.189i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-0.871 + 2.68i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (3.51 + 2.55i)T + (8.96 + 27.5i)T^{2} \)
37 \( 1 - 2.36T + 37T^{2} \)
41 \( 1 + (4.56 + 3.31i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (-5.84 - 4.24i)T + (13.2 + 40.8i)T^{2} \)
47 \( 1 + (-6.83 + 4.96i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-1.25 + 3.84i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-4.48 + 3.25i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 - 12.0T + 61T^{2} \)
67 \( 1 - 10.7T + 67T^{2} \)
71 \( 1 + (-2.26 + 6.96i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (1.54 + 4.74i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (1.72 - 5.29i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (2.25 + 1.63i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (1.46 + 4.49i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (2.06 + 6.34i)T + (-78.4 + 57.0i)T^{2} \)
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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

−9.918067254066127251630485764732, −9.462340332244904749157219015242, −8.477693559931443049160873155187, −7.26591383142632669577297907067, −6.80344864838579405425424887009, −5.54557093307452913843564671158, −4.57816941127317523261944533188, −3.54734467847413854785211421903, −2.25204062676003263024850988946, −0.77277123078316267929624636857, 1.12791020451072619384022934070, 2.42469937613752647610905008373, 3.89474963724154432898868021104, 5.48395229058110349457903399363, 5.84588942469552409941098570460, 6.68450602182892723142451650375, 7.67924371374863493139258926118, 8.539338971028933890194361619909, 9.245698822949870746549113844539, 10.09901301965547762428124844895

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