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Properties

Label 2-104-8.5-c1-0-6
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  + (1.36 + 0.366i)2-s + 2i·3-s + (1.73 + i)4-s − 3.46i·5-s + (−0.732 + 2.73i)6-s − 4.73·7-s + (1.99 + 2i)8-s − 9-s + (1.26 − 4.73i)10-s − 1.26i·11-s + (−2 + 3.46i)12-s + i·13-s + (−6.46 − 1.73i)14-s + 6.92·15-s + (1.99 + 3.46i)16-s − 1.46·17-s + ⋯
L(s)  = 1  + (0.965 + 0.258i)2-s + 1.15i·3-s + (0.866 + 0.5i)4-s − 1.54i·5-s + (−0.298 + 1.11i)6-s − 1.78·7-s + (0.707 + 0.707i)8-s − 0.333·9-s + (0.400 − 1.49i)10-s − 0.382i·11-s + (−0.577 + 0.999i)12-s + 0.277i·13-s + (−1.72 − 0.462i)14-s + 1.78·15-s + (0.499 + 0.866i)16-s − 0.355·17-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

\(L(1)\) \(\approx\) \(1.41988 + 0.588134i\)
\(L(\frac12)\) \(\approx\) \(1.41988 + 0.588134i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.36 - 0.366i)T \)
13 \( 1 - iT \)
good3 \( 1 - 2iT - 3T^{2} \)
5 \( 1 + 3.46iT - 5T^{2} \)
7 \( 1 + 4.73T + 7T^{2} \)
11 \( 1 + 1.26iT - 11T^{2} \)
17 \( 1 + 1.46T + 17T^{2} \)
19 \( 1 + 2.73iT - 19T^{2} \)
23 \( 1 - 4T + 23T^{2} \)
29 \( 1 - 2iT - 29T^{2} \)
31 \( 1 + 3.26T + 31T^{2} \)
37 \( 1 - 4.92iT - 37T^{2} \)
41 \( 1 + 4.92T + 41T^{2} \)
43 \( 1 + 7.46iT - 43T^{2} \)
47 \( 1 - 3.26T + 47T^{2} \)
53 \( 1 - 10.9iT - 53T^{2} \)
59 \( 1 - 0.196iT - 59T^{2} \)
61 \( 1 + 10.9iT - 61T^{2} \)
67 \( 1 + 2.73iT - 67T^{2} \)
71 \( 1 - 2.19T + 71T^{2} \)
73 \( 1 + 0.535T + 73T^{2} \)
79 \( 1 + 1.46T + 79T^{2} \)
83 \( 1 - 6.73iT - 83T^{2} \)
89 \( 1 - 17.3T + 89T^{2} \)
97 \( 1 + 14.3T + 97T^{2} \)
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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

−13.66713348460551342477738804601, −12.97108562948306335020693325602, −12.21653871489260166528855857102, −10.78546377761365198318805053420, −9.514710332505391917145492678906, −8.769027114441600982185064524032, −6.85944620926091183156489416071, −5.50205270731119407441896095195, −4.48177866106527543337411071211, −3.35667813713256326653348392359, 2.48106307219566836217102578034, 3.56327089894437307225813162307, 6.03341198059096965809599785503, 6.75796848113682246252332617157, 7.37029897698755686362765248658, 9.789178435230406437411541607628, 10.64765648977790656680617448684, 11.88890447864364030441797403380, 12.88038802413954090975969281922, 13.36245056849728059608579069507

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