Properties

Label 2-930-93.68-c1-0-15
Degree $2$
Conductor $930$
Sign $0.726 + 0.687i$
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 + (−1.67 − 0.447i)3-s − 4-s + (0.866 + 0.5i)5-s + (−0.447 + 1.67i)6-s + (−1.32 − 2.29i)7-s + i·8-s + (2.59 + 1.49i)9-s + (0.5 − 0.866i)10-s + (−1.35 + 2.34i)11-s + (1.67 + 0.447i)12-s + (3.49 + 2.01i)13-s + (−2.29 + 1.32i)14-s + (−1.22 − 1.22i)15-s + 16-s + (0.370 + 0.642i)17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.966 − 0.258i)3-s − 0.5·4-s + (0.387 + 0.223i)5-s + (−0.182 + 0.683i)6-s + (−0.499 − 0.865i)7-s + 0.353i·8-s + (0.866 + 0.499i)9-s + (0.158 − 0.273i)10-s + (−0.408 + 0.707i)11-s + (0.483 + 0.129i)12-s + (0.970 + 0.560i)13-s + (−0.612 + 0.353i)14-s + (−0.316 − 0.316i)15-s + 0.250·16-s + (0.0899 + 0.155i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.726 + 0.687i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.726 + 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.726 + 0.687i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ 0.726 + 0.687i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.00001 - 0.398261i\)
\(L(\frac12)\) \(\approx\) \(1.00001 - 0.398261i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.67 + 0.447i)T \)
5 \( 1 + (-0.866 - 0.5i)T \)
31 \( 1 + (-4.10 - 3.76i)T \)
good7 \( 1 + (1.32 + 2.29i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.35 - 2.34i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.49 - 2.01i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.370 - 0.642i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.17 - 2.03i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 3.17T + 23T^{2} \)
29 \( 1 - 1.11T + 29T^{2} \)
37 \( 1 + (-8.66 + 5.00i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.56 - 2.63i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.20 + 2.42i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 3.29iT - 47T^{2} \)
53 \( 1 + (-0.432 + 0.748i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.0673 - 0.0388i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + 8.21iT - 61T^{2} \)
67 \( 1 + (-4.09 + 7.08i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-11.6 - 6.71i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-9.24 - 5.33i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (10.5 - 6.09i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.44 - 4.24i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 17.1T + 89T^{2} \)
97 \( 1 - 12.4T + 97T^{2} \)
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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

−10.01855988596137372516830803845, −9.670086078755443211957867645490, −8.268114074048193059035150169564, −7.28786659581080303952313275279, −6.46897533089441681039824984885, −5.65954830079175673924031271697, −4.50753200084704326518488372511, −3.71807542261291705239662082042, −2.18098936577402158282023310730, −0.961494501262997563542538751165, 0.813791942182303643307683866154, 2.85020315841011904328772266603, 4.17598625133437558532233914476, 5.24118560928966836311319724063, 6.04119358016147205432499240348, 6.22751893710422589747733892292, 7.60014244466465996780309654775, 8.513427013952566064425394157213, 9.353123175471362674684795612183, 10.03628853671505025758010289263

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