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Properties

Label 2-930-93.26-c1-0-16
Degree $2$
Conductor $930$
Sign $0.292 + 0.956i$
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  − i·2-s + (−1.23 + 1.21i)3-s − 4-s + (−0.866 + 0.5i)5-s + (1.21 + 1.23i)6-s + (−1.74 + 3.01i)7-s + i·8-s + (0.0529 − 2.99i)9-s + (0.5 + 0.866i)10-s + (−2.28 − 3.95i)11-s + (1.23 − 1.21i)12-s + (−6.16 + 3.55i)13-s + (3.01 + 1.74i)14-s + (0.463 − 1.66i)15-s + 16-s + (1.01 − 1.75i)17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.713 + 0.700i)3-s − 0.5·4-s + (−0.387 + 0.223i)5-s + (0.495 + 0.504i)6-s + (−0.658 + 1.14i)7-s + 0.353i·8-s + (0.0176 − 0.999i)9-s + (0.158 + 0.273i)10-s + (−0.688 − 1.19i)11-s + (0.356 − 0.350i)12-s + (−1.70 + 0.986i)13-s + (0.806 + 0.465i)14-s + (0.119 − 0.430i)15-s + 0.250·16-s + (0.245 − 0.424i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.292 + 0.956i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.292 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.292 + 0.956i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (491, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ 0.292 + 0.956i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.491363 - 0.363349i\)
\(L(\frac12)\) \(\approx\) \(0.491363 - 0.363349i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 + (1.23 - 1.21i)T \)
5 \( 1 + (0.866 - 0.5i)T \)
31 \( 1 + (1.29 - 5.41i)T \)
good7 \( 1 + (1.74 - 3.01i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.28 + 3.95i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (6.16 - 3.55i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.01 + 1.75i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.87 + 6.71i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 7.42T + 23T^{2} \)
29 \( 1 - 6.35T + 29T^{2} \)
37 \( 1 + (-7.71 - 4.45i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.17 + 1.25i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.21 + 1.85i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 5.93iT - 47T^{2} \)
53 \( 1 + (5.09 + 8.83i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (7.20 + 4.16i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 - 2.03iT - 61T^{2} \)
67 \( 1 + (3.08 + 5.35i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-2.26 + 1.30i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-7.81 + 4.51i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (7.33 + 4.23i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.54 + 7.87i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 3.74T + 89T^{2} \)
97 \( 1 - 5.16T + 97T^{2} \)
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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

−9.868151420400190098893875224267, −9.292621223271242061153631347577, −8.665844141860875127344778674211, −7.22273486698269146030756821247, −6.35930096419933971620186685477, −5.04475672377930672509252694985, −4.90170520468890799437316148814, −3.14109121440103278410435320166, −2.79128130407919129831987270895, −0.41662481620707706268946811866, 0.951603918393096521729565256177, 2.83503203183642642740635196444, 4.34988311799746136238733948581, 5.07695273088457389419461419412, 5.98791901679728891009363278329, 7.11215835017821231827400822902, 7.56445855768328424949881330578, 7.943424130085051324306326487381, 9.681185190666609049512351737743, 10.07461368208818088509133301158

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