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Properties

Label 2-40e2-16.5-c1-0-33
Degree $2$
Conductor $1600$
Sign $-0.635 + 0.772i$
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  + (1.86 − 1.86i)3-s − 3.61i·7-s − 3.92i·9-s + (0.0947 + 0.0947i)11-s + (2.59 − 2.59i)13-s − 1.89·17-s + (−2.16 + 2.16i)19-s + (−6.72 − 6.72i)21-s − 5.08i·23-s + (−1.71 − 1.71i)27-s + (−1.25 + 1.25i)29-s + 1.27·31-s + 0.352·33-s + (2.25 + 2.25i)37-s − 9.65i·39-s + ⋯
L(s)  = 1  + (1.07 − 1.07i)3-s − 1.36i·7-s − 1.30i·9-s + (0.0285 + 0.0285i)11-s + (0.719 − 0.719i)13-s − 0.460·17-s + (−0.496 + 0.496i)19-s + (−1.46 − 1.46i)21-s − 1.05i·23-s + (−0.329 − 0.329i)27-s + (−0.233 + 0.233i)29-s + 0.228·31-s + 0.0613·33-s + (0.370 + 0.370i)37-s − 1.54i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.635 + 0.772i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.635 + 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $-0.635 + 0.772i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (1201, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ -0.635 + 0.772i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.331094415\)
\(L(\frac12)\) \(\approx\) \(2.331094415\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.86 + 1.86i)T - 3iT^{2} \)
7 \( 1 + 3.61iT - 7T^{2} \)
11 \( 1 + (-0.0947 - 0.0947i)T + 11iT^{2} \)
13 \( 1 + (-2.59 + 2.59i)T - 13iT^{2} \)
17 \( 1 + 1.89T + 17T^{2} \)
19 \( 1 + (2.16 - 2.16i)T - 19iT^{2} \)
23 \( 1 + 5.08iT - 23T^{2} \)
29 \( 1 + (1.25 - 1.25i)T - 29iT^{2} \)
31 \( 1 - 1.27T + 31T^{2} \)
37 \( 1 + (-2.25 - 2.25i)T + 37iT^{2} \)
41 \( 1 - 8.52iT - 41T^{2} \)
43 \( 1 + (-1.61 - 1.61i)T + 43iT^{2} \)
47 \( 1 + 2.53T + 47T^{2} \)
53 \( 1 + (5.67 + 5.67i)T + 53iT^{2} \)
59 \( 1 + (7.81 + 7.81i)T + 59iT^{2} \)
61 \( 1 + (-3.46 + 3.46i)T - 61iT^{2} \)
67 \( 1 + (-6.29 + 6.29i)T - 67iT^{2} \)
71 \( 1 + 11.3iT - 71T^{2} \)
73 \( 1 - 16.1iT - 73T^{2} \)
79 \( 1 - 1.13T + 79T^{2} \)
83 \( 1 + (-3.75 + 3.75i)T - 83iT^{2} \)
89 \( 1 - 3.98iT - 89T^{2} \)
97 \( 1 - 10.3T + 97T^{2} \)
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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

−8.891885520713166883958052677435, −7.986903939821302541545849491229, −7.86697121323612420508221693673, −6.70125225536470607828270233033, −6.36117484003893541270542728399, −4.80041312450595074437890660954, −3.79910946749699890399262443497, −2.99927178052028319870556913382, −1.82728726406816144662071575090, −0.78777018921078102387308634973, 1.96606731845064066530493811716, 2.78801504351890196515082375707, 3.75151161426074254940324521027, 4.51682510724199742673654434947, 5.51319545843485267634184789435, 6.34065394481654014119284814105, 7.52208345945432247787916294876, 8.519883027702907987937736916503, 9.000149197922108370646109648182, 9.337048474218802665737928723616

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