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Properties

Label 2-930-31.2-c1-0-3
Degree $2$
Conductor $930$
Sign $-0.927 - 0.374i$
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 + (1.10 + 3.40i)7-s + (−0.309 + 0.951i)8-s + (0.309 − 0.951i)9-s + (−0.809 − 0.587i)10-s + (−0.154 − 0.476i)11-s + (−0.809 − 0.587i)12-s + (1.95 − 1.42i)13-s + (−1.10 + 3.40i)14-s + (0.809 − 0.587i)15-s + (−0.809 + 0.587i)16-s + (−2.03 + 6.25i)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 + (0.418 + 1.28i)7-s + (−0.109 + 0.336i)8-s + (0.103 − 0.317i)9-s + (−0.255 − 0.185i)10-s + (−0.0467 − 0.143i)11-s + (−0.233 − 0.169i)12-s + (0.542 − 0.393i)13-s + (−0.295 + 0.910i)14-s + (0.208 − 0.151i)15-s + (−0.202 + 0.146i)16-s + (−0.492 + 1.51i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.927 - 0.374i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.927 - 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.927 - 0.374i$
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: \((2,\ 930,\ (\ :1/2),\ -0.927 - 0.374i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.259543 + 1.33466i\)
\(L(\frac12)\) \(\approx\) \(0.259543 + 1.33466i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.01 + 4.67i)T \)
good7 \( 1 + (-1.10 - 3.40i)T + (-5.66 + 4.11i)T^{2} \)
11 \( 1 + (0.154 + 0.476i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (-1.95 + 1.42i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (2.03 - 6.25i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (2.25 + 1.63i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (2.06 - 6.35i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (-0.171 - 0.124i)T + (8.96 + 27.5i)T^{2} \)
37 \( 1 + 3.80T + 37T^{2} \)
41 \( 1 + (-3.46 - 2.52i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (6.00 + 4.36i)T + (13.2 + 40.8i)T^{2} \)
47 \( 1 + (6.29 - 4.57i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-2.30 + 7.09i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (12.1 - 8.79i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 - 12.6T + 61T^{2} \)
67 \( 1 - 7.80T + 67T^{2} \)
71 \( 1 + (3.02 - 9.31i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-1.47 - 4.55i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (0.461 - 1.42i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-14.2 - 10.3i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (2.94 + 9.06i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-3.57 - 11.0i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.69887496015468103342910700059, −9.498446023795543148823469280794, −8.512153649449546754171588766910, −8.084120132605708323600425074108, −6.80484444792034458967020019369, −5.87708987026887028650704531088, −5.39998738477290681665257160886, −4.27686005890737169392316431480, −3.41860107361872139685570737675, −1.97992598202064107958823241318, 0.55342813394187682421325555469, 1.92646033575689965792127264468, 3.42010534760529194399846405287, 4.42989876146462448489518566292, 4.97794198893190305612034161769, 6.38487136811728404305702421607, 6.97969803908073435815941665790, 7.82862449810101449925093045568, 8.862142176597303857330700764037, 10.05745732135682896165272023023

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