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Properties

Label 2-930-15.2-c1-0-24
Degree $2$
Conductor $930$
Sign $0.847 + 0.531i$
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 + (−0.860 + 1.50i)3-s − 1.00i·4-s + (−2.19 + 0.424i)5-s + (0.454 + 1.67i)6-s + (−0.358 − 0.358i)7-s + (−0.707 − 0.707i)8-s + (−1.52 − 2.58i)9-s + (−1.25 + 1.85i)10-s + 1.95i·11-s + (1.50 + 0.860i)12-s + (1.74 − 1.74i)13-s − 0.507·14-s + (1.24 − 3.66i)15-s − 1.00·16-s + (3.65 − 3.65i)17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (−0.496 + 0.867i)3-s − 0.500i·4-s + (−0.981 + 0.189i)5-s + (0.185 + 0.682i)6-s + (−0.135 − 0.135i)7-s + (−0.250 − 0.250i)8-s + (−0.506 − 0.862i)9-s + (−0.395 + 0.585i)10-s + 0.589i·11-s + (0.433 + 0.248i)12-s + (0.482 − 0.482i)13-s − 0.135·14-s + (0.322 − 0.946i)15-s − 0.250·16-s + (0.887 − 0.887i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.847 + 0.531i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.847 + 0.531i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.847 + 0.531i$
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: \((2,\ 930,\ (\ :1/2),\ 0.847 + 0.531i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.33063 - 0.382954i\)
\(L(\frac12)\) \(\approx\) \(1.33063 - 0.382954i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.860 - 1.50i)T \)
5 \( 1 + (2.19 - 0.424i)T \)
31 \( 1 + T \)
good7 \( 1 + (0.358 + 0.358i)T + 7iT^{2} \)
11 \( 1 - 1.95iT - 11T^{2} \)
13 \( 1 + (-1.74 + 1.74i)T - 13iT^{2} \)
17 \( 1 + (-3.65 + 3.65i)T - 17iT^{2} \)
19 \( 1 + 3.69iT - 19T^{2} \)
23 \( 1 + (-6.27 - 6.27i)T + 23iT^{2} \)
29 \( 1 - 8.51T + 29T^{2} \)
37 \( 1 + (1.92 + 1.92i)T + 37iT^{2} \)
41 \( 1 - 9.77iT - 41T^{2} \)
43 \( 1 + (1.51 - 1.51i)T - 43iT^{2} \)
47 \( 1 + (-4.14 + 4.14i)T - 47iT^{2} \)
53 \( 1 + (9.34 + 9.34i)T + 53iT^{2} \)
59 \( 1 - 1.87T + 59T^{2} \)
61 \( 1 - 10.5T + 61T^{2} \)
67 \( 1 + (7.86 + 7.86i)T + 67iT^{2} \)
71 \( 1 - 2.88iT - 71T^{2} \)
73 \( 1 + (-9.70 + 9.70i)T - 73iT^{2} \)
79 \( 1 + 1.35iT - 79T^{2} \)
83 \( 1 + (-2.46 - 2.46i)T + 83iT^{2} \)
89 \( 1 + 0.192T + 89T^{2} \)
97 \( 1 + (-1.01 - 1.01i)T + 97iT^{2} \)
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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

−10.11332266659837215591186038167, −9.472020264418399555338028496982, −8.458633737086569719499809783186, −7.30445063111038597465722687751, −6.49154378796368830011115126148, −5.17167135418606453193576216650, −4.75158762399638577385767361641, −3.54921789154285824719688656069, −3.02458051125106845218852831966, −0.78442131759677839859011033103, 1.06646080234896642074550162243, 2.86510131504584895047020820916, 3.94992908350975113426938165180, 4.99361814466512290228582251592, 5.97016319856792015265837059298, 6.65027224637449628125399862729, 7.51399747629372722976583583008, 8.334707141322300818910051429607, 8.780542064500239789510703420709, 10.49778760417672479395359580528

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