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Properties

Label 2-930-155.92-c1-0-26
Degree $2$
Conductor $930$
Sign $-0.983 - 0.179i$
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 + (−0.707 − 0.707i)3-s + 1.00i·4-s + (1.94 − 1.09i)5-s + 1.00i·6-s + (−1.54 − 1.54i)7-s + (0.707 − 0.707i)8-s + 1.00i·9-s + (−2.15 − 0.602i)10-s − 1.86i·11-s + (0.707 − 0.707i)12-s + (−1.01 − 1.01i)13-s + 2.18i·14-s + (−2.15 − 0.602i)15-s − 1.00·16-s + (−4.64 + 4.64i)17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (−0.408 − 0.408i)3-s + 0.500i·4-s + (0.871 − 0.490i)5-s + 0.408i·6-s + (−0.583 − 0.583i)7-s + (0.250 − 0.250i)8-s + 0.333i·9-s + (−0.680 − 0.190i)10-s − 0.563i·11-s + (0.204 − 0.204i)12-s + (−0.281 − 0.281i)13-s + 0.583i·14-s + (−0.555 − 0.155i)15-s − 0.250·16-s + (−1.12 + 1.12i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 - 0.179i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.983 - 0.179i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.983 - 0.179i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (247, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ -0.983 - 0.179i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0557534 + 0.615157i\)
\(L(\frac12)\) \(\approx\) \(0.0557534 + 0.615157i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 + (-1.94 + 1.09i)T \)
31 \( 1 + (-5.27 + 1.77i)T \)
good7 \( 1 + (1.54 + 1.54i)T + 7iT^{2} \)
11 \( 1 + 1.86iT - 11T^{2} \)
13 \( 1 + (1.01 + 1.01i)T + 13iT^{2} \)
17 \( 1 + (4.64 - 4.64i)T - 17iT^{2} \)
19 \( 1 + 3.65iT - 19T^{2} \)
23 \( 1 + (2.46 + 2.46i)T + 23iT^{2} \)
29 \( 1 - 3.22T + 29T^{2} \)
37 \( 1 + (5.38 - 5.38i)T - 37iT^{2} \)
41 \( 1 + 5.91T + 41T^{2} \)
43 \( 1 + (3.51 + 3.51i)T + 43iT^{2} \)
47 \( 1 + (5.19 + 5.19i)T + 47iT^{2} \)
53 \( 1 + (7.72 + 7.72i)T + 53iT^{2} \)
59 \( 1 - 4.31iT - 59T^{2} \)
61 \( 1 - 9.74iT - 61T^{2} \)
67 \( 1 + (-7.73 - 7.73i)T + 67iT^{2} \)
71 \( 1 + 0.778T + 71T^{2} \)
73 \( 1 + (-1.92 - 1.92i)T + 73iT^{2} \)
79 \( 1 + 4.71T + 79T^{2} \)
83 \( 1 + (0.710 + 0.710i)T + 83iT^{2} \)
89 \( 1 + 7.40T + 89T^{2} \)
97 \( 1 + (3.69 + 3.69i)T + 97iT^{2} \)
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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

−9.966775331521269051296387126934, −8.650197936164758062547806980475, −8.336772774460639694593352366339, −6.81959090572454605576367918991, −6.45539478368026949300728616911, −5.25699110170823267840524625771, −4.20257468705558764005936302506, −2.83892092058190882837823570603, −1.68630586495341530701126786757, −0.34526117524647954448112360933, 1.87606489449887525654688466743, 3.04771627622104517362893177806, 4.62190218867939333153565718523, 5.44621722873713302136469578015, 6.42898649658104017952507495285, 6.79715199852807479502730074624, 8.001029179623119467655330534391, 9.208563569531201178416605788340, 9.557744526376904730568177245337, 10.24304542739611185006699557821

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