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Properties

Label 2-930-155.123-c1-0-24
Degree $2$
Conductor $930$
Sign $-0.384 + 0.923i$
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  + (0.707 − 0.707i)2-s + (−0.707 + 0.707i)3-s − 1.00i·4-s + (0.0490 − 2.23i)5-s + 1.00i·6-s + (3.32 − 3.32i)7-s + (−0.707 − 0.707i)8-s − 1.00i·9-s + (−1.54 − 1.61i)10-s + 4.17i·11-s + (0.707 + 0.707i)12-s + (2.60 − 2.60i)13-s − 4.70i·14-s + (1.54 + 1.61i)15-s − 1.00·16-s + (−4.75 − 4.75i)17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (−0.408 + 0.408i)3-s − 0.500i·4-s + (0.0219 − 0.999i)5-s + 0.408i·6-s + (1.25 − 1.25i)7-s + (−0.250 − 0.250i)8-s − 0.333i·9-s + (−0.488 − 0.510i)10-s + 1.25i·11-s + (0.204 + 0.204i)12-s + (0.722 − 0.722i)13-s − 1.25i·14-s + (0.399 + 0.417i)15-s − 0.250·16-s + (−1.15 − 1.15i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.384 + 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.384 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.384 + 0.923i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ -0.384 + 0.923i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.06420 - 1.59627i\)
\(L(\frac12)\) \(\approx\) \(1.06420 - 1.59627i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 0.707i)T \)
3 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 + (-0.0490 + 2.23i)T \)
31 \( 1 + (1.00 + 5.47i)T \)
good7 \( 1 + (-3.32 + 3.32i)T - 7iT^{2} \)
11 \( 1 - 4.17iT - 11T^{2} \)
13 \( 1 + (-2.60 + 2.60i)T - 13iT^{2} \)
17 \( 1 + (4.75 + 4.75i)T + 17iT^{2} \)
19 \( 1 - 7.18iT - 19T^{2} \)
23 \( 1 + (-4.10 + 4.10i)T - 23iT^{2} \)
29 \( 1 + 6.16T + 29T^{2} \)
37 \( 1 + (-3.18 - 3.18i)T + 37iT^{2} \)
41 \( 1 - 8.63T + 41T^{2} \)
43 \( 1 + (0.501 - 0.501i)T - 43iT^{2} \)
47 \( 1 + (-0.354 + 0.354i)T - 47iT^{2} \)
53 \( 1 + (5.87 - 5.87i)T - 53iT^{2} \)
59 \( 1 + 2.54iT - 59T^{2} \)
61 \( 1 + 5.63iT - 61T^{2} \)
67 \( 1 + (-7.48 + 7.48i)T - 67iT^{2} \)
71 \( 1 + 13.4T + 71T^{2} \)
73 \( 1 + (-3.37 + 3.37i)T - 73iT^{2} \)
79 \( 1 - 9.67T + 79T^{2} \)
83 \( 1 + (6.43 - 6.43i)T - 83iT^{2} \)
89 \( 1 - 3.82T + 89T^{2} \)
97 \( 1 + (4.24 - 4.24i)T - 97iT^{2} \)
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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

−9.938379309319203219409065060063, −9.223707478171593838982067232821, −8.062187782064885252646426927179, −7.37182909292093305322493968708, −6.07747185018620937038853955855, −4.98203567816627701123682594822, −4.54178595195636877721245325854, −3.82854839034638168395192912747, −1.96357873364632260912839050108, −0.848483179815316595252089783623, 1.85786163695460757967983483636, 2.92552818755470311402961183770, 4.21707102274862567183192219733, 5.37714553440997012063118663122, 5.99826743651648179752727949066, 6.74950385851289934561155551500, 7.64414090570030193365101817882, 8.685985604226026119515411644817, 9.020503698899926205231037665916, 11.02569706646446820380554261609

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