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Properties

Label 2-930-15.2-c1-0-40
Degree $2$
Conductor $930$
Sign $0.685 + 0.727i$
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 + (1.22 + 1.22i)3-s − 1.00i·4-s + (−0.448 − 2.19i)5-s + 1.73·6-s + (2 + 2i)7-s + (−0.707 − 0.707i)8-s + 2.99i·9-s + (−1.86 − 1.23i)10-s − 4.76i·11-s + (1.22 − 1.22i)12-s + (1.63 − 1.63i)13-s + 2.82·14-s + (2.13 − 3.23i)15-s − 1.00·16-s + (0.707 − 0.707i)17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (0.707 + 0.707i)3-s − 0.500i·4-s + (−0.200 − 0.979i)5-s + 0.707·6-s + (0.755 + 0.755i)7-s + (−0.250 − 0.250i)8-s + 0.999i·9-s + (−0.590 − 0.389i)10-s − 1.43i·11-s + (0.353 − 0.353i)12-s + (0.453 − 0.453i)13-s + 0.755·14-s + (0.550 − 0.834i)15-s − 0.250·16-s + (0.171 − 0.171i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.685 + 0.727i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.685 + 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.685 + 0.727i$
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: \((2,\ 930,\ (\ :1/2),\ 0.685 + 0.727i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.51983 - 1.08827i\)
\(L(\frac12)\) \(\approx\) \(2.51983 - 1.08827i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.22 - 1.22i)T \)
5 \( 1 + (0.448 + 2.19i)T \)
31 \( 1 + T \)
good7 \( 1 + (-2 - 2i)T + 7iT^{2} \)
11 \( 1 + 4.76iT - 11T^{2} \)
13 \( 1 + (-1.63 + 1.63i)T - 13iT^{2} \)
17 \( 1 + (-0.707 + 0.707i)T - 17iT^{2} \)
19 \( 1 + 3iT - 19T^{2} \)
23 \( 1 + (-2.44 - 2.44i)T + 23iT^{2} \)
29 \( 1 - 9.52T + 29T^{2} \)
37 \( 1 + (-6.46 - 6.46i)T + 37iT^{2} \)
41 \( 1 + 5.65iT - 41T^{2} \)
43 \( 1 + (3.46 - 3.46i)T - 43iT^{2} \)
47 \( 1 + (2.63 - 2.63i)T - 47iT^{2} \)
53 \( 1 + (4.89 + 4.89i)T + 53iT^{2} \)
59 \( 1 + 4.62T + 59T^{2} \)
61 \( 1 + 14.1T + 61T^{2} \)
67 \( 1 + (-8.83 - 8.83i)T + 67iT^{2} \)
71 \( 1 + 3.06iT - 71T^{2} \)
73 \( 1 + (2.53 - 2.53i)T - 73iT^{2} \)
79 \( 1 + 0.267iT - 79T^{2} \)
83 \( 1 + (-6.88 - 6.88i)T + 83iT^{2} \)
89 \( 1 - 9.89T + 89T^{2} \)
97 \( 1 + (6.63 + 6.63i)T + 97iT^{2} \)
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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

−9.912002100426748361456315066028, −9.006635175057119480451304794593, −8.498411482779996320269947107730, −7.890316361406680035426482108786, −6.14574825905516083772294626806, −5.18077674395035086217153030472, −4.71519536927802565687467383308, −3.51696839423593325724463480947, −2.69224575136385437313890772517, −1.19735092118363966755840722823, 1.64234347350837857110472166922, 2.80868022738189471318854129617, 3.93316079955597113027209424657, 4.67331010642742842905684484650, 6.27847963059607183434910141948, 6.80123231809883651843451111012, 7.67186793066810742051753982166, 7.991954896145750896573880201723, 9.204753965165266202084772599127, 10.20475867115363924399737570366

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