Properties

Label 2-40e2-20.3-c1-0-14
Degree $2$
Conductor $1600$
Sign $0.850 - 0.525i$
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 − i)3-s + (3 + 3i)7-s + i·9-s − 2i·11-s + (3 + 3i)13-s + (−1 + i)17-s − 4·19-s + 6·21-s + (1 − i)23-s + (4 + 4i)27-s + 10i·31-s + (−2 − 2i)33-s + (−1 + i)37-s + 6·39-s − 10·41-s + ⋯
L(s)  = 1  + (0.577 − 0.577i)3-s + (1.13 + 1.13i)7-s + 0.333i·9-s − 0.603i·11-s + (0.832 + 0.832i)13-s + (−0.242 + 0.242i)17-s − 0.917·19-s + 1.30·21-s + (0.208 − 0.208i)23-s + (0.769 + 0.769i)27-s + 1.79i·31-s + (−0.348 − 0.348i)33-s + (−0.164 + 0.164i)37-s + 0.960·39-s − 1.56·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $0.850 - 0.525i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (1343, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ 0.850 - 0.525i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.356187041\)
\(L(\frac12)\) \(\approx\) \(2.356187041\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 + i)T - 3iT^{2} \)
7 \( 1 + (-3 - 3i)T + 7iT^{2} \)
11 \( 1 + 2iT - 11T^{2} \)
13 \( 1 + (-3 - 3i)T + 13iT^{2} \)
17 \( 1 + (1 - i)T - 17iT^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 + (-1 + i)T - 23iT^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 - 10iT - 31T^{2} \)
37 \( 1 + (1 - i)T - 37iT^{2} \)
41 \( 1 + 10T + 41T^{2} \)
43 \( 1 + (-5 + 5i)T - 43iT^{2} \)
47 \( 1 + (-3 - 3i)T + 47iT^{2} \)
53 \( 1 + (5 + 5i)T + 53iT^{2} \)
59 \( 1 - 12T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 + (1 + i)T + 67iT^{2} \)
71 \( 1 - 2iT - 71T^{2} \)
73 \( 1 + (1 + i)T + 73iT^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 + (-5 + 5i)T - 83iT^{2} \)
89 \( 1 + 16iT - 89T^{2} \)
97 \( 1 + (-3 + 3i)T - 97iT^{2} \)
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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

−8.949308776173010957975980837711, −8.615438685139658252791054915969, −8.217952852428854810315473712050, −7.11841544533514631319757746441, −6.31477005881578345439531718352, −5.35193605369363449701521096422, −4.57260612362092408740704908740, −3.30927135895031105960132761393, −2.17910011568493508616522677277, −1.56039085313696789733358049108, 0.911043372697545569805261962514, 2.24730671604403008144562798173, 3.58089537743508553096294215741, 4.18423420983815244687014448918, 4.94092230917738424743200509192, 6.11097635043789657226929159402, 7.07947089281660569271320211210, 7.925309679445865231984154554998, 8.454380564982761455140661602334, 9.363258489822003334166133824785

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