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Properties

Label 2-40e2-80.67-c1-0-25
Degree $2$
Conductor $1600$
Sign $-0.586 + 0.810i$
Analytic cond. $12.7760$
Root an. cond. $3.57436$
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  − 0.790i·3-s + (0.139 + 0.139i)7-s + 2.37·9-s + (−2.94 − 2.94i)11-s + 0.235·13-s + (−2.06 − 2.06i)17-s + (−2.55 − 2.55i)19-s + (0.110 − 0.110i)21-s + (−4.62 + 4.62i)23-s − 4.24i·27-s + (6.66 − 6.66i)29-s − 3.43i·31-s + (−2.32 + 2.32i)33-s − 1.38·37-s − 0.186i·39-s + ⋯
L(s)  = 1  − 0.456i·3-s + (0.0528 + 0.0528i)7-s + 0.791·9-s + (−0.888 − 0.888i)11-s + 0.0653·13-s + (−0.499 − 0.499i)17-s + (−0.585 − 0.585i)19-s + (0.0241 − 0.0241i)21-s + (−0.964 + 0.964i)23-s − 0.817i·27-s + (1.23 − 1.23i)29-s − 0.616i·31-s + (−0.405 + 0.405i)33-s − 0.227·37-s − 0.0298i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.586 + 0.810i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.586 + 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $-0.586 + 0.810i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (1007, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ -0.586 + 0.810i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.176283932\)
\(L(\frac12)\) \(\approx\) \(1.176283932\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good3 \( 1 + 0.790iT - 3T^{2} \)
7 \( 1 + (-0.139 - 0.139i)T + 7iT^{2} \)
11 \( 1 + (2.94 + 2.94i)T + 11iT^{2} \)
13 \( 1 - 0.235T + 13T^{2} \)
17 \( 1 + (2.06 + 2.06i)T + 17iT^{2} \)
19 \( 1 + (2.55 + 2.55i)T + 19iT^{2} \)
23 \( 1 + (4.62 - 4.62i)T - 23iT^{2} \)
29 \( 1 + (-6.66 + 6.66i)T - 29iT^{2} \)
31 \( 1 + 3.43iT - 31T^{2} \)
37 \( 1 + 1.38T + 37T^{2} \)
41 \( 1 - 8.26iT - 41T^{2} \)
43 \( 1 + 5.40T + 43T^{2} \)
47 \( 1 + (-6.84 + 6.84i)T - 47iT^{2} \)
53 \( 1 - 8.19iT - 53T^{2} \)
59 \( 1 + (-4.32 + 4.32i)T - 59iT^{2} \)
61 \( 1 + (9.15 + 9.15i)T + 61iT^{2} \)
67 \( 1 + 5.00T + 67T^{2} \)
71 \( 1 - 6.06T + 71T^{2} \)
73 \( 1 + (11.3 + 11.3i)T + 73iT^{2} \)
79 \( 1 - 4.44T + 79T^{2} \)
83 \( 1 + 11.4iT - 83T^{2} \)
89 \( 1 + 5.84T + 89T^{2} \)
97 \( 1 + (-0.515 - 0.515i)T + 97iT^{2} \)
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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

−9.089707587265540997992515509121, −8.143148968890229284897249705787, −7.67812742779848652973224979453, −6.66716438513406427268798878882, −6.01935333956031476438883259992, −4.96909662542296311715411668812, −4.10984998758335016340675160587, −2.90317479523802500118904796270, −1.91459738364381739634771306818, −0.44296119480138737094203867546, 1.58815658130115384634621036247, 2.68193340382926393633569479211, 4.01816632397841987725461033303, 4.55637624289783721385478641105, 5.47069770392125532188309699865, 6.57310787900050605753138267784, 7.23909016154600866563380595884, 8.187303735837762412990711085588, 8.855457734478144016746951042320, 9.883718742132905238628868991508

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