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Properties

Label 2-930-15.2-c1-0-53
Degree $2$
Conductor $930$
Sign $-0.915 - 0.401i$
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 + (−1.72 − 0.0980i)3-s − 1.00i·4-s + (0.278 − 2.21i)5-s + (−1.29 + 1.15i)6-s + (−1.59 − 1.59i)7-s + (−0.707 − 0.707i)8-s + (2.98 + 0.339i)9-s + (−1.37 − 1.76i)10-s − 3.70i·11-s + (−0.0980 + 1.72i)12-s + (−1.02 + 1.02i)13-s − 2.25·14-s + (−0.699 + 3.80i)15-s − 1.00·16-s + (−0.479 + 0.479i)17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (−0.998 − 0.0566i)3-s − 0.500i·4-s + (0.124 − 0.992i)5-s + (−0.527 + 0.470i)6-s + (−0.602 − 0.602i)7-s + (−0.250 − 0.250i)8-s + (0.993 + 0.113i)9-s + (−0.433 − 0.558i)10-s − 1.11i·11-s + (−0.0283 + 0.499i)12-s + (−0.284 + 0.284i)13-s − 0.602·14-s + (−0.180 + 0.983i)15-s − 0.250·16-s + (−0.116 + 0.116i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.915 - 0.401i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.915 - 0.401i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.915 - 0.401i$
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: \((2,\ 930,\ (\ :1/2),\ -0.915 - 0.401i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.170143 + 0.811235i\)
\(L(\frac12)\) \(\approx\) \(0.170143 + 0.811235i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.72 + 0.0980i)T \)
5 \( 1 + (-0.278 + 2.21i)T \)
31 \( 1 + T \)
good7 \( 1 + (1.59 + 1.59i)T + 7iT^{2} \)
11 \( 1 + 3.70iT - 11T^{2} \)
13 \( 1 + (1.02 - 1.02i)T - 13iT^{2} \)
17 \( 1 + (0.479 - 0.479i)T - 17iT^{2} \)
19 \( 1 + 0.296iT - 19T^{2} \)
23 \( 1 + (-1.49 - 1.49i)T + 23iT^{2} \)
29 \( 1 - 1.30T + 29T^{2} \)
37 \( 1 + (2.24 + 2.24i)T + 37iT^{2} \)
41 \( 1 - 5.01iT - 41T^{2} \)
43 \( 1 + (7.37 - 7.37i)T - 43iT^{2} \)
47 \( 1 + (-0.230 + 0.230i)T - 47iT^{2} \)
53 \( 1 + (-0.282 - 0.282i)T + 53iT^{2} \)
59 \( 1 + 12.3T + 59T^{2} \)
61 \( 1 - 11.1T + 61T^{2} \)
67 \( 1 + (9.42 + 9.42i)T + 67iT^{2} \)
71 \( 1 + 6.43iT - 71T^{2} \)
73 \( 1 + (5.91 - 5.91i)T - 73iT^{2} \)
79 \( 1 + 3.56iT - 79T^{2} \)
83 \( 1 + (-1.95 - 1.95i)T + 83iT^{2} \)
89 \( 1 - 10.7T + 89T^{2} \)
97 \( 1 + (-5.06 - 5.06i)T + 97iT^{2} \)
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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

−9.790477023526471257572188085585, −9.013545172247771700625371351745, −7.85766351252408270675048256311, −6.69567827462687730847740277622, −5.98461889144452420498062508476, −5.11622953578999011302847924665, −4.34247265572184129186017817192, −3.31079106702625779600253886010, −1.54263332323718047352910953923, −0.37459128064546090714190523170, 2.19160604865721739638404998826, 3.39802456179961496670287992858, 4.56627032800621835090888419090, 5.46099749028746720716654761962, 6.26882551057614501620465552953, 6.95537385587084672501322945516, 7.53079495741756975841162937228, 8.945327860176716115818875069351, 9.964403923815003605017438408225, 10.43737206296361420478401040475

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