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Properties

Label 2-104-8.5-c1-0-1
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  + (−0.366 − 1.36i)2-s + 2i·3-s + (−1.73 + i)4-s + 3.46i·5-s + (2.73 − 0.732i)6-s − 1.26·7-s + (2 + 1.99i)8-s − 9-s + (4.73 − 1.26i)10-s − 4.73i·11-s + (−2 − 3.46i)12-s + i·13-s + (0.464 + 1.73i)14-s − 6.92·15-s + (1.99 − 3.46i)16-s + 5.46·17-s + ⋯
L(s)  = 1  + (−0.258 − 0.965i)2-s + 1.15i·3-s + (−0.866 + 0.5i)4-s + 1.54i·5-s + (1.11 − 0.298i)6-s − 0.479·7-s + (0.707 + 0.707i)8-s − 0.333·9-s + (1.49 − 0.400i)10-s − 1.42i·11-s + (−0.577 − 0.999i)12-s + 0.277i·13-s + (0.124 + 0.462i)14-s − 1.78·15-s + (0.499 − 0.866i)16-s + 1.32·17-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

\(L(1)\) \(\approx\) \(0.756322 + 0.313279i\)
\(L(\frac12)\) \(\approx\) \(0.756322 + 0.313279i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.366 + 1.36i)T \)
13 \( 1 - iT \)
good3 \( 1 - 2iT - 3T^{2} \)
5 \( 1 - 3.46iT - 5T^{2} \)
7 \( 1 + 1.26T + 7T^{2} \)
11 \( 1 + 4.73iT - 11T^{2} \)
17 \( 1 - 5.46T + 17T^{2} \)
19 \( 1 - 0.732iT - 19T^{2} \)
23 \( 1 - 4T + 23T^{2} \)
29 \( 1 - 2iT - 29T^{2} \)
31 \( 1 + 6.73T + 31T^{2} \)
37 \( 1 + 8.92iT - 37T^{2} \)
41 \( 1 - 8.92T + 41T^{2} \)
43 \( 1 + 0.535iT - 43T^{2} \)
47 \( 1 - 6.73T + 47T^{2} \)
53 \( 1 + 2.92iT - 53T^{2} \)
59 \( 1 + 10.1iT - 59T^{2} \)
61 \( 1 - 2.92iT - 61T^{2} \)
67 \( 1 - 0.732iT - 67T^{2} \)
71 \( 1 + 8.19T + 71T^{2} \)
73 \( 1 + 7.46T + 73T^{2} \)
79 \( 1 - 5.46T + 79T^{2} \)
83 \( 1 - 3.26iT - 83T^{2} \)
89 \( 1 + 17.3T + 89T^{2} \)
97 \( 1 - 6.39T + 97T^{2} \)
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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

−14.13138761130749281911969637587, −12.74559147605110047710515783750, −11.22936593423537180748242896467, −10.80347344142736103279795786413, −9.927458358212439716120657493433, −8.999722199462404836567638118216, −7.39930033665639993270289510149, −5.65706916745960590564750799286, −3.75203968154316350580012571200, −3.06320050117788563455232785257, 1.23000020128735635730991750156, 4.56746430097494365243897227587, 5.76050998433423880724887136900, 7.15368383405723015950202175585, 7.88710400559734548488988068574, 9.093944912592433757917183976042, 9.975598714293025556706726442622, 12.22471881712449308840253616255, 12.77478134770588876714157957757, 13.41041458260834501861777971835

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