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Properties

Label 2-40e2-80.69-c1-0-3
Degree $2$
Conductor $1600$
Sign $-0.998 + 0.0496i$
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  + (2.16 + 2.16i)3-s − 3.30·7-s + 6.40i·9-s + (−2.01 − 2.01i)11-s + (−0.794 − 0.794i)13-s + 4.61i·17-s + (−3.48 + 3.48i)19-s + (−7.16 − 7.16i)21-s − 7.99·23-s + (−7.38 + 7.38i)27-s + (1.95 − 1.95i)29-s + 5.12·31-s − 8.72i·33-s + (−0.448 + 0.448i)37-s − 3.44i·39-s + ⋯
L(s)  = 1  + (1.25 + 1.25i)3-s − 1.24·7-s + 2.13i·9-s + (−0.606 − 0.606i)11-s + (−0.220 − 0.220i)13-s + 1.11i·17-s + (−0.800 + 0.800i)19-s + (−1.56 − 1.56i)21-s − 1.66·23-s + (−1.42 + 1.42i)27-s + (0.362 − 0.362i)29-s + 0.920·31-s − 1.51i·33-s + (−0.0736 + 0.0736i)37-s − 0.551i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0496i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0496i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $-0.998 + 0.0496i$
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: \((2,\ 1600,\ (\ :1/2),\ -0.998 + 0.0496i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.148459546\)
\(L(\frac12)\) \(\approx\) \(1.148459546\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.16 - 2.16i)T + 3iT^{2} \)
7 \( 1 + 3.30T + 7T^{2} \)
11 \( 1 + (2.01 + 2.01i)T + 11iT^{2} \)
13 \( 1 + (0.794 + 0.794i)T + 13iT^{2} \)
17 \( 1 - 4.61iT - 17T^{2} \)
19 \( 1 + (3.48 - 3.48i)T - 19iT^{2} \)
23 \( 1 + 7.99T + 23T^{2} \)
29 \( 1 + (-1.95 + 1.95i)T - 29iT^{2} \)
31 \( 1 - 5.12T + 31T^{2} \)
37 \( 1 + (0.448 - 0.448i)T - 37iT^{2} \)
41 \( 1 + 4.02iT - 41T^{2} \)
43 \( 1 + (4.97 - 4.97i)T - 43iT^{2} \)
47 \( 1 - 5.49iT - 47T^{2} \)
53 \( 1 + (3.35 - 3.35i)T - 53iT^{2} \)
59 \( 1 + (-2.07 - 2.07i)T + 59iT^{2} \)
61 \( 1 + (0.557 - 0.557i)T - 61iT^{2} \)
67 \( 1 + (-0.636 - 0.636i)T + 67iT^{2} \)
71 \( 1 + 6.85iT - 71T^{2} \)
73 \( 1 - 10.5T + 73T^{2} \)
79 \( 1 + 17.3T + 79T^{2} \)
83 \( 1 + (-9.48 - 9.48i)T + 83iT^{2} \)
89 \( 1 - 7.62iT - 89T^{2} \)
97 \( 1 - 0.709iT - 97T^{2} \)
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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

−9.965427703268700925934628640366, −9.083703006598493433445338002341, −8.179930328298932081589301604703, −7.979503436316831726573294438857, −6.43152430532156044624720521410, −5.76493332752456422968043784361, −4.49900647429831196011507906561, −3.77071684240764791981653747262, −3.10234411104987442655742248171, −2.18980816910076260825881116346, 0.33708375924044294898250289078, 2.01675140941942830762297463908, 2.68888488479086327140892801430, 3.51430793975512793433395983426, 4.72810002781432538085944514384, 6.14513499282022923869434536699, 6.81656416149061869273304849903, 7.32815495328704294753514939619, 8.211864147061651291725467032515, 8.851518638690800178250704056134

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