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Properties

Label 2-104-8.5-c1-0-3
Degree $2$
Conductor $104$
Sign $0.707 - 0.707i$
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  + (−1 + i)2-s − i·3-s − 2i·4-s + 3i·5-s + (1 + i)6-s + 3·7-s + (2 + 2i)8-s + 2·9-s + (−3 − 3i)10-s − 2·12-s + i·13-s + (−3 + 3i)14-s + 3·15-s − 4·16-s − 7·17-s + (−2 + 2i)18-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s − 0.577i·3-s − i·4-s + 1.34i·5-s + (0.408 + 0.408i)6-s + 1.13·7-s + (0.707 + 0.707i)8-s + 0.666·9-s + (−0.948 − 0.948i)10-s − 0.577·12-s + 0.277i·13-s + (−0.801 + 0.801i)14-s + 0.774·15-s − 16-s − 1.69·17-s + (−0.471 + 0.471i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(104\)    =    \(2^{3} \cdot 13\)
Sign: $0.707 - 0.707i$
Analytic conductor: \(0.830444\)
Root analytic conductor: \(0.911287\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{104} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 104,\ (\ :1/2),\ 0.707 - 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.752271 + 0.311601i\)
\(L(\frac12)\) \(\approx\) \(0.752271 + 0.311601i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 - i)T \)
13 \( 1 - iT \)
good3 \( 1 + iT - 3T^{2} \)
5 \( 1 - 3iT - 5T^{2} \)
7 \( 1 - 3T + 7T^{2} \)
11 \( 1 - 11T^{2} \)
17 \( 1 + 7T + 17T^{2} \)
19 \( 1 + 4iT - 19T^{2} \)
23 \( 1 - 4T + 23T^{2} \)
29 \( 1 + 4iT - 29T^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 - 7iT - 37T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 + iT - 43T^{2} \)
47 \( 1 + 7T + 47T^{2} \)
53 \( 1 - 4iT - 53T^{2} \)
59 \( 1 + 14iT - 59T^{2} \)
61 \( 1 + 10iT - 61T^{2} \)
67 \( 1 - 2iT - 67T^{2} \)
71 \( 1 + 3T + 71T^{2} \)
73 \( 1 - 14T + 73T^{2} \)
79 \( 1 + 10T + 79T^{2} \)
83 \( 1 - 14iT - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 8T + 97T^{2} \)
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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

−14.15440631000278160403879527824, −13.20852039438558746320617634653, −11.26399418836677676300121220421, −10.92856591718551450502520099306, −9.508812300858634453042338551359, −8.195138470375305827482347337470, −7.10015539971666652880095457705, −6.57035023944590742165974568824, −4.73879432561462994763240805629, −2.05759140072052388639569224374, 1.61690266206879990138128123661, 4.10530874863142447259873660113, 5.01634651797727751766763907520, 7.40053747850704823055861792041, 8.626171664997409426871552911051, 9.199662015815668273650387336959, 10.52597930620038220776880637397, 11.34323865242704371941958808037, 12.58600300582480593983015104864, 13.19648486565889657619060078782

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