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Properties

Label 2-40e2-80.69-c1-0-19
Degree $2$
Conductor $1600$
Sign $0.755 - 0.655i$
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  + (1.15 + 1.15i)3-s + 4.31·7-s − 0.316i·9-s + (0.158 + 0.158i)11-s + (2.31 + 2.31i)13-s + 5.31i·17-s + (3.15 − 3.15i)19-s + (5 + 5i)21-s − 6.31·23-s + (3.84 − 3.84i)27-s + (−2 + 2i)29-s − 4.31·31-s + 0.366i·33-s + (7.31 − 7.31i)37-s + 5.36i·39-s + ⋯
L(s)  = 1  + (0.668 + 0.668i)3-s + 1.63·7-s − 0.105i·9-s + (0.0477 + 0.0477i)11-s + (0.642 + 0.642i)13-s + 1.28i·17-s + (0.724 − 0.724i)19-s + (1.09 + 1.09i)21-s − 1.31·23-s + (0.739 − 0.739i)27-s + (−0.371 + 0.371i)29-s − 0.775·31-s + 0.0638i·33-s + (1.20 − 1.20i)37-s + 0.859i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 - 0.655i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.755 - 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $0.755 - 0.655i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ 0.755 - 0.655i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.701091661\)
\(L(\frac12)\) \(\approx\) \(2.701091661\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.15 - 1.15i)T + 3iT^{2} \)
7 \( 1 - 4.31T + 7T^{2} \)
11 \( 1 + (-0.158 - 0.158i)T + 11iT^{2} \)
13 \( 1 + (-2.31 - 2.31i)T + 13iT^{2} \)
17 \( 1 - 5.31iT - 17T^{2} \)
19 \( 1 + (-3.15 + 3.15i)T - 19iT^{2} \)
23 \( 1 + 6.31T + 23T^{2} \)
29 \( 1 + (2 - 2i)T - 29iT^{2} \)
31 \( 1 + 4.31T + 31T^{2} \)
37 \( 1 + (-7.31 + 7.31i)T - 37iT^{2} \)
41 \( 1 - 5iT - 41T^{2} \)
43 \( 1 + (-5.63 + 5.63i)T - 43iT^{2} \)
47 \( 1 + 8iT - 47T^{2} \)
53 \( 1 + (3.31 - 3.31i)T - 53iT^{2} \)
59 \( 1 + (-5.31 - 5.31i)T + 59iT^{2} \)
61 \( 1 + (3.63 - 3.63i)T - 61iT^{2} \)
67 \( 1 + (-5.84 - 5.84i)T + 67iT^{2} \)
71 \( 1 + 4.63iT - 71T^{2} \)
73 \( 1 + 13.3T + 73T^{2} \)
79 \( 1 + 2.31T + 79T^{2} \)
83 \( 1 + (-3.84 - 3.84i)T + 83iT^{2} \)
89 \( 1 - 15.9iT - 89T^{2} \)
97 \( 1 + 6.63iT - 97T^{2} \)
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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

−9.295247260457208885604562214250, −8.753453799690906184266763694863, −8.087552908401588315611730402569, −7.33567933907308108251240284627, −6.14596907532387278943790532689, −5.29126517669093065261099997843, −4.15505184972610506359712914653, −3.89016405922814762309906997739, −2.40421156003738987700176575446, −1.39046080465949488503208775173, 1.18460582449770691323952190763, 2.06025729453734920668323104117, 3.09220046309466853982638166169, 4.32127614573152083966022450577, 5.20125697458044302799151088010, 6.00624229340979897494582146208, 7.29344595432980245360040381489, 7.927812137160315386690357859700, 8.138539672164265496701823753467, 9.154987478861236518388529317051

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