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Properties

Label 2-104-104.77-c1-0-9
Degree $2$
Conductor $104$
Sign $0.889 + 0.456i$
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  + (1.29 + 0.569i)2-s − 2.94i·3-s + (1.35 + 1.47i)4-s − 1.81·5-s + (1.68 − 3.81i)6-s + 1.13i·7-s + (0.908 + 2.67i)8-s − 5.70·9-s + (−2.35 − 1.03i)10-s + 4.40·11-s + (4.35 − 3.98i)12-s + (−2.58 + 2.50i)13-s + (−0.649 + 1.47i)14-s + 5.35i·15-s + (−0.350 + 3.98i)16-s + 0.701·17-s + ⋯
L(s)  = 1  + (0.915 + 0.402i)2-s − 1.70i·3-s + (0.675 + 0.737i)4-s − 0.812·5-s + (0.686 − 1.55i)6-s + 0.430i·7-s + (0.321 + 0.947i)8-s − 1.90·9-s + (−0.743 − 0.327i)10-s + 1.32·11-s + (1.25 − 1.15i)12-s + (−0.717 + 0.696i)13-s + (−0.173 + 0.394i)14-s + 1.38i·15-s + (−0.0876 + 0.996i)16-s + 0.170·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.889 + 0.456i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.889 + 0.456i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(104\)    =    \(2^{3} \cdot 13\)
Sign: $0.889 + 0.456i$
Analytic conductor: \(0.830444\)
Root analytic conductor: \(0.911287\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{104} (77, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 104,\ (\ :1/2),\ 0.889 + 0.456i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.43383 - 0.346373i\)
\(L(\frac12)\) \(\approx\) \(1.43383 - 0.346373i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.29 - 0.569i)T \)
13 \( 1 + (2.58 - 2.50i)T \)
good3 \( 1 + 2.94iT - 3T^{2} \)
5 \( 1 + 1.81T + 5T^{2} \)
7 \( 1 - 1.13iT - 7T^{2} \)
11 \( 1 - 4.40T + 11T^{2} \)
17 \( 1 - 0.701T + 17T^{2} \)
19 \( 1 + 5.95T + 19T^{2} \)
23 \( 1 - 4T + 23T^{2} \)
29 \( 1 + 5.01iT - 29T^{2} \)
31 \( 1 + 8.77iT - 31T^{2} \)
37 \( 1 + 3.36T + 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 2.94iT - 43T^{2} \)
47 \( 1 - 1.13iT - 47T^{2} \)
53 \( 1 - 11.7iT - 53T^{2} \)
59 \( 1 - 5.95T + 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 7.49T + 67T^{2} \)
71 \( 1 + 11.8iT - 71T^{2} \)
73 \( 1 - 8.43iT - 73T^{2} \)
79 \( 1 + 10.8T + 79T^{2} \)
83 \( 1 - 2.85T + 83T^{2} \)
89 \( 1 + 8.43iT - 89T^{2} \)
97 \( 1 - 12.9iT - 97T^{2} \)
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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

−13.64040132774767893662901093211, −12.62807264874181927289716880071, −11.93814791030391414329399615599, −11.40447568252851811032545090701, −8.843229019152769318145069847715, −7.74867168339589962282225778393, −6.89152293933785278497367764155, −6.02173935562708062804357498065, −4.13969252270409509403080934438, −2.23840596654893780815658189644, 3.37044467719642366778055732738, 4.19353017314670095559826744363, 5.19491961205994334633081774837, 6.86788814341503763582096916372, 8.734624862051932070877751649536, 9.986277011557524305115789662086, 10.76290119999745627952998854052, 11.62545318407273436793430961356, 12.67170540198096834792133938919, 14.31591179291197013255862957959

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