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Properties

Label 2-1040-65.18-c1-0-30
Degree $2$
Conductor $1040$
Sign $0.256 + 0.966i$
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  + (1 + i)3-s + (−2 − i)5-s + 2·7-s − i·9-s + (−1 − i)11-s + (2 − 3i)13-s + (−1 − 3i)15-s + (−5 − 5i)17-s + (−3 − 3i)19-s + (2 + 2i)21-s + (−5 + 5i)23-s + (3 + 4i)25-s + (4 − 4i)27-s − 4i·29-s + (−1 + i)31-s + ⋯
L(s)  = 1  + (0.577 + 0.577i)3-s + (−0.894 − 0.447i)5-s + 0.755·7-s − 0.333i·9-s + (−0.301 − 0.301i)11-s + (0.554 − 0.832i)13-s + (−0.258 − 0.774i)15-s + (−1.21 − 1.21i)17-s + (−0.688 − 0.688i)19-s + (0.436 + 0.436i)21-s + (−1.04 + 1.04i)23-s + (0.600 + 0.800i)25-s + (0.769 − 0.769i)27-s − 0.742i·29-s + (−0.179 + 0.179i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.256 + 0.966i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.256 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1040\)    =    \(2^{4} \cdot 5 \cdot 13\)
Sign: $0.256 + 0.966i$
Analytic conductor: \(8.30444\)
Root analytic conductor: \(2.88174\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1040} (993, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1040,\ (\ :1/2),\ 0.256 + 0.966i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.401197730\)
\(L(\frac12)\) \(\approx\) \(1.401197730\)
\(L(\frac{3}{2})\) 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 \( 1 + (2 + i)T \)
13 \( 1 + (-2 + 3i)T \)
good3 \( 1 + (-1 - i)T + 3iT^{2} \)
7 \( 1 - 2T + 7T^{2} \)
11 \( 1 + (1 + i)T + 11iT^{2} \)
17 \( 1 + (5 + 5i)T + 17iT^{2} \)
19 \( 1 + (3 + 3i)T + 19iT^{2} \)
23 \( 1 + (5 - 5i)T - 23iT^{2} \)
29 \( 1 + 4iT - 29T^{2} \)
31 \( 1 + (1 - i)T - 31iT^{2} \)
37 \( 1 - 8T + 37T^{2} \)
41 \( 1 + (-1 + i)T - 41iT^{2} \)
43 \( 1 + (-5 + 5i)T - 43iT^{2} \)
47 \( 1 - 2T + 47T^{2} \)
53 \( 1 + (1 + i)T + 53iT^{2} \)
59 \( 1 + (-3 + 3i)T - 59iT^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 + 12iT - 67T^{2} \)
71 \( 1 + (1 - i)T - 71iT^{2} \)
73 \( 1 + 6iT - 73T^{2} \)
79 \( 1 - 14iT - 79T^{2} \)
83 \( 1 - 6T + 83T^{2} \)
89 \( 1 + (-7 + 7i)T - 89iT^{2} \)
97 \( 1 - 2iT - 97T^{2} \)
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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

−9.509806481208583062154229351161, −8.915435379981158077061093524589, −8.157738623794823279394018742334, −7.58196642151275347870376363467, −6.36290452529170314907378797674, −5.16811363488569300287350594687, −4.35511126917189943872673111175, −3.60169524973038943669873108975, −2.45336088730345454751686095044, −0.58729887436870695833733660103, 1.70954616237536616600080804234, 2.54589447621572045048106582646, 4.04631720974902632468588002048, 4.51094663839515464586977036084, 6.09143063926719895443603856660, 6.85921458646353935523354443674, 7.82780794040564635443276285170, 8.261264591062637742581030157993, 8.892113506858774627922244029088, 10.34691257935545990430446847593

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