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Properties

Label 2-1040-20.7-c1-0-34
Degree $2$
Conductor $1040$
Sign $-0.998 + 0.0618i$
Analytic cond. $8.30444$
Root an. cond. $2.88174$
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  + (−0.611 − 0.611i)3-s + (0.509 − 2.17i)5-s + (−1.48 + 1.48i)7-s − 2.25i·9-s − 2.63i·11-s + (−0.707 + 0.707i)13-s + (−1.64 + 1.01i)15-s + (0.501 + 0.501i)17-s − 5.25·19-s + 1.81·21-s + (2.39 + 2.39i)23-s + (−4.48 − 2.21i)25-s + (−3.20 + 3.20i)27-s − 6.31i·29-s + 3.70i·31-s + ⋯
L(s)  = 1  + (−0.352 − 0.352i)3-s + (0.227 − 0.973i)5-s + (−0.561 + 0.561i)7-s − 0.751i·9-s − 0.795i·11-s + (−0.196 + 0.196i)13-s + (−0.423 + 0.263i)15-s + (0.121 + 0.121i)17-s − 1.20·19-s + 0.396·21-s + (0.499 + 0.499i)23-s + (−0.896 − 0.443i)25-s + (−0.617 + 0.617i)27-s − 1.17i·29-s + 0.665i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0618i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0618i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1040\)    =    \(2^{4} \cdot 5 \cdot 13\)
Sign: $-0.998 + 0.0618i$
Analytic conductor: \(8.30444\)
Root analytic conductor: \(2.88174\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1040} (287, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1040,\ (\ :1/2),\ -0.998 + 0.0618i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6121397700\)
\(L(\frac12)\) \(\approx\) \(0.6121397700\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-0.509 + 2.17i)T \)
13 \( 1 + (0.707 - 0.707i)T \)
good3 \( 1 + (0.611 + 0.611i)T + 3iT^{2} \)
7 \( 1 + (1.48 - 1.48i)T - 7iT^{2} \)
11 \( 1 + 2.63iT - 11T^{2} \)
17 \( 1 + (-0.501 - 0.501i)T + 17iT^{2} \)
19 \( 1 + 5.25T + 19T^{2} \)
23 \( 1 + (-2.39 - 2.39i)T + 23iT^{2} \)
29 \( 1 + 6.31iT - 29T^{2} \)
31 \( 1 - 3.70iT - 31T^{2} \)
37 \( 1 + (0.738 + 0.738i)T + 37iT^{2} \)
41 \( 1 + 6.68T + 41T^{2} \)
43 \( 1 + (2.31 + 2.31i)T + 43iT^{2} \)
47 \( 1 + (2.31 - 2.31i)T - 47iT^{2} \)
53 \( 1 + (-5.12 + 5.12i)T - 53iT^{2} \)
59 \( 1 + 8.79T + 59T^{2} \)
61 \( 1 + 5.44T + 61T^{2} \)
67 \( 1 + (0.136 - 0.136i)T - 67iT^{2} \)
71 \( 1 + 7.29iT - 71T^{2} \)
73 \( 1 + (-1.75 + 1.75i)T - 73iT^{2} \)
79 \( 1 + 15.6T + 79T^{2} \)
83 \( 1 + (-5.29 - 5.29i)T + 83iT^{2} \)
89 \( 1 - 7.06iT - 89T^{2} \)
97 \( 1 + (1.06 + 1.06i)T + 97iT^{2} \)
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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

−9.329054331582823689434871397101, −8.830126076381420396276468741833, −8.013904427313922042388882304625, −6.74183909848661783560165520968, −6.08993133832756501557504493209, −5.38867402753666063700350890088, −4.24276204905224047919090490072, −3.12224778684578792672173911663, −1.67942099442008636853781422599, −0.27572018247623768144446989679, 2.00271537471334386567993622294, 3.09191079440635985193016783016, 4.22239460745716047622087533624, 5.10378064678174864175166843918, 6.20668404170685844186686589388, 6.95773927071127983160575811271, 7.61240236935671286341383242734, 8.744106805118155364332060636775, 9.886630730166294393894505226095, 10.32642294903207807539145430703

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