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Properties

Label 2-104-13.12-c1-0-3
Degree $2$
Conductor $104$
Sign $-0.155 + 0.987i$
Analytic cond. $0.830444$
Root an. cond. $0.911287$
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  − 2.56·3-s − 2.56i·5-s − 4.56i·7-s + 3.56·9-s + 3.12i·11-s + (−3.56 − 0.561i)13-s + 6.56i·15-s + 0.561·17-s − 2i·19-s + 11.6i·21-s + 5.12·23-s − 1.56·25-s − 1.43·27-s + 3.12·29-s − 3.12i·31-s + ⋯
L(s)  = 1  − 1.47·3-s − 1.14i·5-s − 1.72i·7-s + 1.18·9-s + 0.941i·11-s + (−0.987 − 0.155i)13-s + 1.69i·15-s + 0.136·17-s − 0.458i·19-s + 2.54i·21-s + 1.06·23-s − 0.312·25-s − 0.276·27-s + 0.579·29-s − 0.560i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.155 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.155 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(104\)    =    \(2^{3} \cdot 13\)
Sign: $-0.155 + 0.987i$
Analytic conductor: \(0.830444\)
Root analytic conductor: \(0.911287\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{104} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 104,\ (\ :1/2),\ -0.155 + 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.375143 - 0.438926i\)
\(L(\frac12)\) \(\approx\) \(0.375143 - 0.438926i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 + (3.56 + 0.561i)T \)
good3 \( 1 + 2.56T + 3T^{2} \)
5 \( 1 + 2.56iT - 5T^{2} \)
7 \( 1 + 4.56iT - 7T^{2} \)
11 \( 1 - 3.12iT - 11T^{2} \)
17 \( 1 - 0.561T + 17T^{2} \)
19 \( 1 + 2iT - 19T^{2} \)
23 \( 1 - 5.12T + 23T^{2} \)
29 \( 1 - 3.12T + 29T^{2} \)
31 \( 1 + 3.12iT - 31T^{2} \)
37 \( 1 + 5.43iT - 37T^{2} \)
41 \( 1 - 8iT - 41T^{2} \)
43 \( 1 - 5.43T + 43T^{2} \)
47 \( 1 - 3.43iT - 47T^{2} \)
53 \( 1 + 4.24T + 53T^{2} \)
59 \( 1 + 10iT - 59T^{2} \)
61 \( 1 - 11.1T + 61T^{2} \)
67 \( 1 - 0.876iT - 67T^{2} \)
71 \( 1 - 5.68iT - 71T^{2} \)
73 \( 1 + 15.3iT - 73T^{2} \)
79 \( 1 + 2.87T + 79T^{2} \)
83 \( 1 - 8.24iT - 83T^{2} \)
89 \( 1 + 5.12iT - 89T^{2} \)
97 \( 1 - 10.2iT - 97T^{2} \)
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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

−13.05788105325865886984199441831, −12.55064043438921653300935219018, −11.39803804061105034109458613702, −10.44408913303524753807604362885, −9.502943919287802888050606754840, −7.65090547894969370312473309833, −6.71524237113646937116762012197, −5.05576853582456256570451100711, −4.45745836243593014118859789520, −0.822306837426084776611752528850, 2.81940109496769999616361812104, 5.17542745090515959892732581799, 6.00595867533401268694237966537, 6.98664653332785130469973438510, 8.717655708598016115979700991346, 10.13985013584147892272852406779, 11.12304775058398219087056879111, 11.84270170950207514109985740900, 12.59468315299087929444429183686, 14.26437492633309436610098698076

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