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Properties

Label 2-104-104.77-c1-0-7
Degree $2$
Conductor $104$
Sign $0.741 + 0.670i$
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  + (0.273 − 1.38i)2-s + 1.51i·3-s + (−1.85 − 0.758i)4-s + 3.11·5-s + (2.10 + 0.414i)6-s − 2.77i·7-s + (−1.55 + 2.36i)8-s + 0.701·9-s + (0.850 − 4.32i)10-s − 2.56·11-s + (1.14 − 2.80i)12-s + (−0.546 + 3.56i)13-s + (−3.85 − 0.758i)14-s + 4.72i·15-s + (2.85 + 2.80i)16-s − 5.70·17-s + ⋯
L(s)  = 1  + (0.193 − 0.981i)2-s + 0.875i·3-s + (−0.925 − 0.379i)4-s + 1.39·5-s + (0.858 + 0.169i)6-s − 1.04i·7-s + (−0.550 + 0.834i)8-s + 0.233·9-s + (0.269 − 1.36i)10-s − 0.774·11-s + (0.331 − 0.809i)12-s + (−0.151 + 0.988i)13-s + (−1.02 − 0.202i)14-s + 1.21i·15-s + (0.712 + 0.701i)16-s − 1.38·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.741 + 0.670i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.741 + 0.670i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(104\)    =    \(2^{3} \cdot 13\)
Sign: $0.741 + 0.670i$
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: \((2,\ 104,\ (\ :1/2),\ 0.741 + 0.670i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.09905 - 0.423251i\)
\(L(\frac12)\) \(\approx\) \(1.09905 - 0.423251i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.273 + 1.38i)T \)
13 \( 1 + (0.546 - 3.56i)T \)
good3 \( 1 - 1.51iT - 3T^{2} \)
5 \( 1 - 3.11T + 5T^{2} \)
7 \( 1 + 2.77iT - 7T^{2} \)
11 \( 1 + 2.56T + 11T^{2} \)
17 \( 1 + 5.70T + 17T^{2} \)
19 \( 1 + 4.75T + 19T^{2} \)
23 \( 1 - 4T + 23T^{2} \)
29 \( 1 + 7.12iT - 29T^{2} \)
31 \( 1 - 3.60iT - 31T^{2} \)
37 \( 1 + 4.20T + 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 - 1.51iT - 43T^{2} \)
47 \( 1 + 2.77iT - 47T^{2} \)
53 \( 1 + 6.06iT - 53T^{2} \)
59 \( 1 - 4.75T + 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 12.0T + 67T^{2} \)
71 \( 1 + 6.66iT - 71T^{2} \)
73 \( 1 - 14.9iT - 73T^{2} \)
79 \( 1 - 14.8T + 79T^{2} \)
83 \( 1 + 9.89T + 83T^{2} \)
89 \( 1 + 14.9iT - 89T^{2} \)
97 \( 1 - 3.89iT - 97T^{2} \)
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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

−13.46776806443947657670497047403, −12.94621800343662273538753424946, −11.16392439531813485315100608515, −10.41233782112278402632283057284, −9.775380239750554482290196874979, −8.808803340393507431798639766953, −6.66602712481839184016848003136, −5.03370194316158520117162790168, −4.09297786652745936322154971920, −2.13772188594754425789548013358, 2.39130313911093480849587290440, 5.07353111568576508110953504175, 6.03003534123826089633134764111, 6.96331048139805700160081798585, 8.341105097231345813654578286918, 9.270217292447693315573833081049, 10.55035433555259352956896908896, 12.57862617510232310901576550869, 12.96964393131648905934262263271, 13.73184040615781609738296468679

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