Properties

Label 2-930-31.25-c1-0-8
Degree $2$
Conductor $930$
Sign $0.275 - 0.961i$
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  + 2-s + (0.5 + 0.866i)3-s + 4-s + (−0.5 + 0.866i)5-s + (0.5 + 0.866i)6-s + (−0.914 − 1.58i)7-s + 8-s + (−0.499 + 0.866i)9-s + (−0.5 + 0.866i)10-s + (−0.914 + 1.58i)11-s + (0.5 + 0.866i)12-s + (−1 + 1.73i)13-s + (−0.914 − 1.58i)14-s − 0.999·15-s + 16-s + (3.82 + 6.63i)17-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.288 + 0.499i)3-s + 0.5·4-s + (−0.223 + 0.387i)5-s + (0.204 + 0.353i)6-s + (−0.345 − 0.598i)7-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (−0.158 + 0.273i)10-s + (−0.275 + 0.477i)11-s + (0.144 + 0.249i)12-s + (−0.277 + 0.480i)13-s + (−0.244 − 0.423i)14-s − 0.258·15-s + 0.250·16-s + (0.928 + 1.60i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.275 - 0.961i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.275 - 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.275 - 0.961i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ 0.275 - 0.961i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.93204 + 1.45681i\)
\(L(\frac12)\) \(\approx\) \(1.93204 + 1.45681i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
31 \( 1 + (3.5 - 4.33i)T \)
good7 \( 1 + (0.914 + 1.58i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.914 - 1.58i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1 - 1.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.82 - 6.63i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3 - 5.19i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 7.65T + 23T^{2} \)
29 \( 1 - 0.171T + 29T^{2} \)
37 \( 1 + (-0.828 - 1.43i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-5.65 + 9.79i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.82 + 8.36i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 13.3T + 47T^{2} \)
53 \( 1 + (-0.671 + 1.16i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.0857 - 0.148i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + 6T + 61T^{2} \)
67 \( 1 + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-6 + 10.3i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (4.65 - 8.06i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.32 + 7.49i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 12T + 89T^{2} \)
97 \( 1 - 5.48T + 97T^{2} \)
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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

−10.37916160016497596804777939312, −9.647380903686874655068437306763, −8.485907005868594123067641353179, −7.53672326073195375052215696316, −6.88098410537339448176642206375, −5.77387490738977905225721319888, −4.86980095929789332818456272714, −3.74390266352252071840969878313, −3.30466956022544491499451502130, −1.76661990927838286419989684950, 0.923206742438276791233764286658, 2.78999455642625949447812630077, 3.12495196303928649145075565161, 4.81511315984825902205024287999, 5.35835415038875783852424626624, 6.42342064341223383446788770427, 7.37654748236167257717657695643, 7.984467108672455078539190702213, 9.187139719282135143140460014519, 9.631141503827898318262933932444

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