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Properties

Label 2-930-31.2-c1-0-15
Degree $2$
Conductor $930$
Sign $0.777 - 0.628i$
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  + (0.809 + 0.587i)2-s + (0.809 − 0.587i)3-s + (0.309 + 0.951i)4-s + 5-s + 6-s + (0.837 + 2.57i)7-s + (−0.309 + 0.951i)8-s + (0.309 − 0.951i)9-s + (0.809 + 0.587i)10-s + (−0.878 − 2.70i)11-s + (0.809 + 0.587i)12-s + (4.10 − 2.98i)13-s + (−0.837 + 2.57i)14-s + (0.809 − 0.587i)15-s + (−0.809 + 0.587i)16-s + (−1.86 + 5.75i)17-s + ⋯
L(s)  = 1  + (0.572 + 0.415i)2-s + (0.467 − 0.339i)3-s + (0.154 + 0.475i)4-s + 0.447·5-s + 0.408·6-s + (0.316 + 0.974i)7-s + (−0.109 + 0.336i)8-s + (0.103 − 0.317i)9-s + (0.255 + 0.185i)10-s + (−0.264 − 0.814i)11-s + (0.233 + 0.169i)12-s + (1.13 − 0.827i)13-s + (−0.223 + 0.689i)14-s + (0.208 − 0.151i)15-s + (−0.202 + 0.146i)16-s + (−0.453 + 1.39i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.777 - 0.628i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.777 - 0.628i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.777 - 0.628i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (901, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ 0.777 - 0.628i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.73182 + 0.966609i\)
\(L(\frac12)\) \(\approx\) \(2.73182 + 0.966609i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.809 - 0.587i)T \)
3 \( 1 + (-0.809 + 0.587i)T \)
5 \( 1 - T \)
31 \( 1 + (-5.55 + 0.370i)T \)
good7 \( 1 + (-0.837 - 2.57i)T + (-5.66 + 4.11i)T^{2} \)
11 \( 1 + (0.878 + 2.70i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (-4.10 + 2.98i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (1.86 - 5.75i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-3.22 - 2.34i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-0.470 + 1.44i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (-0.805 - 0.585i)T + (8.96 + 27.5i)T^{2} \)
37 \( 1 + 10.9T + 37T^{2} \)
41 \( 1 + (-1.57 - 1.14i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (1.38 + 1.00i)T + (13.2 + 40.8i)T^{2} \)
47 \( 1 + (2.71 - 1.97i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (0.127 - 0.393i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (5.37 - 3.90i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + 7.87T + 61T^{2} \)
67 \( 1 - 14.0T + 67T^{2} \)
71 \( 1 + (-4.03 + 12.4i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-0.759 - 2.33i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-2.06 + 6.36i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (11.2 + 8.15i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (4.57 + 14.0i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (1.58 + 4.87i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.24360561103473274600289117216, −8.880632931507934810418659265076, −8.468874462349555626226847873884, −7.79815874162014560858697186614, −6.41557956643566294560575612980, −5.93887249919802341460793646662, −5.12425165442727752563085073572, −3.69725733124244583384689784443, −2.87153657133732979041420390039, −1.59944924271343378892306786088, 1.32283985168626964233040675130, 2.54768062403172060757351727050, 3.68033335377777068868993101791, 4.57830010528003895078472899941, 5.25389391905862755803852653766, 6.68804631624745344627373495767, 7.22080984796469467202139144739, 8.431697304059560327188308449105, 9.421223520092554497476140481824, 9.934455689245478638549941477043

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