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Properties

Label 2-930-31.2-c1-0-4
Degree $2$
Conductor $930$
Sign $-0.634 - 0.773i$
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 + (−0.302 − 0.931i)7-s + (−0.309 + 0.951i)8-s + (0.309 − 0.951i)9-s + (0.809 + 0.587i)10-s + (0.197 + 0.606i)11-s + (−0.809 − 0.587i)12-s + (−5.66 + 4.11i)13-s + (0.302 − 0.931i)14-s + (−0.809 + 0.587i)15-s + (−0.809 + 0.587i)16-s + (−1.21 + 3.75i)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 + (−0.114 − 0.352i)7-s + (−0.109 + 0.336i)8-s + (0.103 − 0.317i)9-s + (0.255 + 0.185i)10-s + (0.0594 + 0.183i)11-s + (−0.233 − 0.169i)12-s + (−1.57 + 1.14i)13-s + (0.0809 − 0.249i)14-s + (−0.208 + 0.151i)15-s + (−0.202 + 0.146i)16-s + (−0.295 + 0.909i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.634 - 0.773i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.634 - 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.634 - 0.773i$
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: \((2,\ 930,\ (\ :1/2),\ -0.634 - 0.773i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.676237 + 1.42966i\)
\(L(\frac12)\) \(\approx\) \(0.676237 + 1.42966i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.409 - 5.55i)T \)
good7 \( 1 + (0.302 + 0.931i)T + (-5.66 + 4.11i)T^{2} \)
11 \( 1 + (-0.197 - 0.606i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (5.66 - 4.11i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (1.21 - 3.75i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-5.17 - 3.76i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (0.319 - 0.982i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (-1.97 - 1.43i)T + (8.96 + 27.5i)T^{2} \)
37 \( 1 - 1.93T + 37T^{2} \)
41 \( 1 + (-0.827 - 0.601i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (4.17 + 3.03i)T + (13.2 + 40.8i)T^{2} \)
47 \( 1 + (5.48 - 3.98i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-0.824 + 2.53i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (4.13 - 3.00i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + 8.93T + 61T^{2} \)
67 \( 1 - 8.74T + 67T^{2} \)
71 \( 1 + (3.07 - 9.45i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (3.76 + 11.5i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-0.0759 + 0.233i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-12.0 - 8.76i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (4.06 + 12.5i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (3.54 + 10.9i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.20834348990566092632582832513, −9.721249703662336631964080853551, −8.737498712919503946303038288234, −7.53122264579827828216054073507, −6.86856191236596458914653398510, −6.01497793958933938388865571606, −5.06531635691892618471666964705, −4.37966353209046123450486457487, −3.26282230418668319313932646081, −1.80306516740310880858847823579, 0.63703250945560425797955386264, 2.34725546900212830277303298763, 3.04479447369667790552025848332, 4.73211850325273424619298605345, 5.25360827139390506689488339445, 6.12840544551680087633764335558, 7.11263533469341664379827844300, 7.87260108399891497111628747480, 9.289032483265408062970453241154, 9.816001987245945441345089802330

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