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Properties

Label 2-1040-65.64-c1-0-37
Degree $2$
Conductor $1040$
Sign $-0.998 + 0.0621i$
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  − 3.09i·3-s + (−1.73 − 1.41i)5-s + 4.37·7-s − 6.58·9-s − 2.53i·11-s + (2.64 − 2.44i)13-s + (−4.37 + 5.36i)15-s + 1.29i·17-s − 5.36i·19-s − 13.5i·21-s − 1.80i·23-s + (0.999 + 4.89i)25-s + 11.0i·27-s − 7.58·29-s − 3.12i·31-s + ⋯
L(s)  = 1  − 1.78i·3-s + (−0.774 − 0.632i)5-s + 1.65·7-s − 2.19·9-s − 0.763i·11-s + (0.733 − 0.679i)13-s + (−1.13 + 1.38i)15-s + 0.313i·17-s − 1.23i·19-s − 2.95i·21-s − 0.376i·23-s + (0.199 + 0.979i)25-s + 2.13i·27-s − 1.40·29-s − 0.561i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0621i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0621i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1040\)    =    \(2^{4} \cdot 5 \cdot 13\)
Sign: $-0.998 + 0.0621i$
Analytic conductor: \(8.30444\)
Root analytic conductor: \(2.88174\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1040} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1040,\ (\ :1/2),\ -0.998 + 0.0621i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.478973514\)
\(L(\frac12)\) \(\approx\) \(1.478973514\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.73 + 1.41i)T \)
13 \( 1 + (-2.64 + 2.44i)T \)
good3 \( 1 + 3.09iT - 3T^{2} \)
7 \( 1 - 4.37T + 7T^{2} \)
11 \( 1 + 2.53iT - 11T^{2} \)
17 \( 1 - 1.29iT - 17T^{2} \)
19 \( 1 + 5.36iT - 19T^{2} \)
23 \( 1 + 1.80iT - 23T^{2} \)
29 \( 1 + 7.58T + 29T^{2} \)
31 \( 1 + 3.12iT - 31T^{2} \)
37 \( 1 - 2.55T + 37T^{2} \)
41 \( 1 - 7.89iT - 41T^{2} \)
43 \( 1 - 4.38iT - 43T^{2} \)
47 \( 1 - 6.20T + 47T^{2} \)
53 \( 1 - 6.19iT - 53T^{2} \)
59 \( 1 - 10.4iT - 59T^{2} \)
61 \( 1 - 3.58T + 61T^{2} \)
67 \( 1 + 2.55T + 67T^{2} \)
71 \( 1 + 0.295iT - 71T^{2} \)
73 \( 1 - 2.55T + 73T^{2} \)
79 \( 1 + 11.1T + 79T^{2} \)
83 \( 1 - 0.723T + 83T^{2} \)
89 \( 1 + 11.3iT - 89T^{2} \)
97 \( 1 - 12.2T + 97T^{2} \)
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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

−8.924994819682110796321070965876, −8.470014658167782281367630839232, −7.78361900378949946588611353225, −7.37316537708266298563407568596, −6.11586419615174774194309788883, −5.38259495595440516344636623306, −4.27502553052422013828277058982, −2.82552218790330637387401457049, −1.53715173120383381371292275265, −0.72017572713467960964056173533, 2.02794997602795997338809477988, 3.62208007578404373301494864667, 4.08102545023462166094297386077, 4.92588912614534331374202265785, 5.71614240711988763824422915131, 7.15695226414944000580348359160, 8.036051929830782549346089248838, 8.726102778592732479079558575703, 9.597106159049414565635797855447, 10.48120165064662304739854933209

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