Properties

Label 2-40e2-8.5-c1-0-12
Degree $2$
Conductor $1600$
Sign $0.258 - 0.965i$
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  + i·3-s + 3.46·7-s + 2·9-s + 3i·11-s + 3.46i·13-s − 3·17-s + i·19-s + 3.46i·21-s + 5i·27-s − 10.3i·29-s + 6.92·31-s − 3·33-s + 10.3i·37-s − 3.46·39-s − 9·41-s + ⋯
L(s)  = 1  + 0.577i·3-s + 1.30·7-s + 0.666·9-s + 0.904i·11-s + 0.960i·13-s − 0.727·17-s + 0.229i·19-s + 0.755i·21-s + 0.962i·27-s − 1.92i·29-s + 1.24·31-s − 0.522·33-s + 1.70i·37-s − 0.554·39-s − 1.40·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 - 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $0.258 - 0.965i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (801, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ 0.258 - 0.965i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.049879322\)
\(L(\frac12)\) \(\approx\) \(2.049879322\)
\(L(\frac{3}{2})\) 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 \( 1 - iT - 3T^{2} \)
7 \( 1 - 3.46T + 7T^{2} \)
11 \( 1 - 3iT - 11T^{2} \)
13 \( 1 - 3.46iT - 13T^{2} \)
17 \( 1 + 3T + 17T^{2} \)
19 \( 1 - iT - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 10.3iT - 29T^{2} \)
31 \( 1 - 6.92T + 31T^{2} \)
37 \( 1 - 10.3iT - 37T^{2} \)
41 \( 1 + 9T + 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 - 10.3T + 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 12iT - 59T^{2} \)
61 \( 1 - 3.46iT - 61T^{2} \)
67 \( 1 - 11iT - 67T^{2} \)
71 \( 1 - 10.3T + 71T^{2} \)
73 \( 1 + 7T + 73T^{2} \)
79 \( 1 + 10.3T + 79T^{2} \)
83 \( 1 - 15iT - 83T^{2} \)
89 \( 1 + 3T + 89T^{2} \)
97 \( 1 + 14T + 97T^{2} \)
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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

−9.823566576200825298969625767688, −8.737073049638742642994390152970, −8.085073982003788169188022837109, −7.17353854376480542809786109343, −6.46067345777764791777940945302, −5.15638641956372212996978496981, −4.47945833697788903812045583287, −4.05116931983957084273570151249, −2.36693742157622901068982187610, −1.47973081456259457162912681181, 0.876262531421629958077602881430, 1.89303465965727662156161528727, 3.08521673887515311622077937819, 4.30987531222875992573776096024, 5.11884345635840674063961106964, 5.98089814944552536976952890433, 7.00304214269075922175671002007, 7.63265843011285257335009493275, 8.425524660694140207033237339247, 8.945334489453855508915989127609

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