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Properties

Label 2-40e2-5.4-c1-0-4
Degree $2$
Conductor $1600$
Sign $0.894 - 0.447i$
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  − 3i·3-s + 2i·7-s − 6·9-s − 11-s + 4i·13-s + 5i·17-s + 19-s + 6·21-s + 2i·23-s + 9i·27-s − 8·29-s + 10·31-s + 3i·33-s + 6i·37-s + 12·39-s + ⋯
L(s)  = 1  − 1.73i·3-s + 0.755i·7-s − 2·9-s − 0.301·11-s + 1.10i·13-s + 1.21i·17-s + 0.229·19-s + 1.30·21-s + 0.417i·23-s + 1.73i·27-s − 1.48·29-s + 1.79·31-s + 0.522i·33-s + 0.986i·37-s + 1.92·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $0.894 - 0.447i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ 0.894 - 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.131097438\)
\(L(\frac12)\) \(\approx\) \(1.131097438\)
\(L(\frac{3}{2})\) 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 \( 1 + 3iT - 3T^{2} \)
7 \( 1 - 2iT - 7T^{2} \)
11 \( 1 + T + 11T^{2} \)
13 \( 1 - 4iT - 13T^{2} \)
17 \( 1 - 5iT - 17T^{2} \)
19 \( 1 - T + 19T^{2} \)
23 \( 1 - 2iT - 23T^{2} \)
29 \( 1 + 8T + 29T^{2} \)
31 \( 1 - 10T + 31T^{2} \)
37 \( 1 - 6iT - 37T^{2} \)
41 \( 1 + 3T + 41T^{2} \)
43 \( 1 - 4iT - 43T^{2} \)
47 \( 1 - 4iT - 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 - 8T + 59T^{2} \)
61 \( 1 + 10T + 61T^{2} \)
67 \( 1 - iT - 67T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 + 3iT - 73T^{2} \)
79 \( 1 + 6T + 79T^{2} \)
83 \( 1 + 13iT - 83T^{2} \)
89 \( 1 - 9T + 89T^{2} \)
97 \( 1 + 14iT - 97T^{2} \)
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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

−9.172233008589101017225964234474, −8.513325630923809575768664163890, −7.81236172682069523212773415653, −7.09745507008846527341507454473, −6.20563951633160571426882134350, −5.82959345824437763961307185915, −4.54389805745358666724377655179, −3.12842516438000579818624841082, −2.12685928326681266783326889312, −1.36799207208325963679370961364, 0.44636699284835547915522807559, 2.66411158964718331863472444671, 3.50108155382465016065228418309, 4.30551214951101528837659676053, 5.12463359720506145016660818974, 5.70906291585543162607006596579, 7.01624557358276738604468088384, 7.894062691419872661851029667603, 8.746598502943036555869545077196, 9.558458105415433116124152083751

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