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Properties

Label 2-1300-13.12-c1-0-13
Degree $2$
Conductor $1300$
Sign $0.936 - 0.349i$
Analytic cond. $10.3805$
Root an. cond. $3.22188$
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  + 2.26·3-s − 1.11i·7-s + 2.11·9-s + 5.37i·11-s + (3.37 − 1.26i)13-s + 7.90i·19-s − 2.52i·21-s + 6.49·23-s − 2·27-s + 3.63·29-s − 3.14i·31-s + 12.1i·33-s − 7.40i·37-s + (7.63 − 2.85i)39-s − 4.75i·41-s + ⋯
L(s)  = 1  + 1.30·3-s − 0.421i·7-s + 0.705·9-s + 1.62i·11-s + (0.936 − 0.349i)13-s + 1.81i·19-s − 0.550i·21-s + 1.35·23-s − 0.384·27-s + 0.675·29-s − 0.565i·31-s + 2.11i·33-s − 1.21i·37-s + (1.22 − 0.456i)39-s − 0.742i·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.936 - 0.349i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.936 - 0.349i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1300\)    =    \(2^{2} \cdot 5^{2} \cdot 13\)
Sign: $0.936 - 0.349i$
Analytic conductor: \(10.3805\)
Root analytic conductor: \(3.22188\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1300} (701, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1300,\ (\ :1/2),\ 0.936 - 0.349i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.703156115\)
\(L(\frac12)\) \(\approx\) \(2.703156115\)
\(L(\frac{3}{2})\) 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 \( 1 \)
13 \( 1 + (-3.37 + 1.26i)T \)
good3 \( 1 - 2.26T + 3T^{2} \)
7 \( 1 + 1.11iT - 7T^{2} \)
11 \( 1 - 5.37iT - 11T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 7.90iT - 19T^{2} \)
23 \( 1 - 6.49T + 23T^{2} \)
29 \( 1 - 3.63T + 29T^{2} \)
31 \( 1 + 3.14iT - 31T^{2} \)
37 \( 1 + 7.40iT - 37T^{2} \)
41 \( 1 + 4.75iT - 41T^{2} \)
43 \( 1 - 4.78T + 43T^{2} \)
47 \( 1 - 6.16iT - 47T^{2} \)
53 \( 1 + 0.292T + 53T^{2} \)
59 \( 1 + 11.3iT - 59T^{2} \)
61 \( 1 + 5.34T + 61T^{2} \)
67 \( 1 + 11.8iT - 67T^{2} \)
71 \( 1 - 3.43iT - 71T^{2} \)
73 \( 1 + 4.59iT - 73T^{2} \)
79 \( 1 + 10.8T + 79T^{2} \)
83 \( 1 - 7.40iT - 83T^{2} \)
89 \( 1 - 14.8iT - 89T^{2} \)
97 \( 1 - 3.76iT - 97T^{2} \)
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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

−9.527838548158398227207407736510, −8.955450760689270691499788229858, −7.947449023192187309777300116913, −7.58292621505233606100968244900, −6.59022287322251126768750735418, −5.46112497844512939164218382735, −4.23436358251629982336924217185, −3.60931151730308059135233494397, −2.49791395813051271948575900244, −1.46780962973521744141122318495, 1.13041634602321394023080616508, 2.78682156848527183777743699715, 3.07244919141925099122868275355, 4.26373039736029013297382389809, 5.41350532466315464069454661563, 6.39208298326251917424009927170, 7.24754220124818419787130229654, 8.372351271638359677923797653533, 8.818217434712672370814181784024, 9.096237821146451052679120285999

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