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Properties

Label 2-104-104.77-c1-0-8
Degree $2$
Conductor $104$
Sign $-0.428 + 0.903i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.428 + 0.903i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.428 + 0.903i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(104\)    =    \(2^{3} \cdot 13\)
Sign: $-0.428 + 0.903i$
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: \((2,\ 104,\ (\ :1/2),\ -0.428 + 0.903i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.397261 - 0.628221i\)
\(L(\frac12)\) \(\approx\) \(0.397261 - 0.628221i\)
\(L(\frac{3}{2})\) 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

−13.40578719536985884843360576000, −12.41289418873654545973867803185, −11.40360335964814989987887202522, −10.29212370559073068303208011063, −8.979003329318700646770803902159, −7.80710362176191074427142421296, −7.05335111708918813821168578481, −5.84464183189456589231510466564, −2.76043834425376321410567890553, −1.33987006982212205472820492609, 2.92775046910224693932946339474, 5.20237187628060806751075860265, 5.80360564955885668765529553562, 7.85531290475141873599149033963, 9.045448012535112797393488484437, 9.809963039592461852947806587266, 10.52992004660716634984024502575, 11.40293752484838568968666272456, 13.33275317940235526223358191708, 14.60346302107533337368234048301

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