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Properties

Label 2-104-8.5-c1-0-9
Degree $2$
Conductor $104$
Sign $0.0864 + 0.996i$
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  + (0.671 − 1.24i)2-s − 0.146i·3-s + (−1.09 − 1.67i)4-s − i·5-s + (−0.182 − 0.0982i)6-s − 0.146·7-s + (−2.81 + 0.244i)8-s + 2.97·9-s + (−1.24 − 0.671i)10-s + 2.68i·11-s + (−0.244 + 0.160i)12-s + i·13-s + (−0.0982 + 0.182i)14-s − 0.146·15-s + (−1.58 + 3.67i)16-s + 17-s + ⋯
L(s)  = 1  + (0.474 − 0.880i)2-s − 0.0845i·3-s + (−0.549 − 0.835i)4-s − 0.447i·5-s + (−0.0743 − 0.0401i)6-s − 0.0553·7-s + (−0.996 + 0.0864i)8-s + 0.992·9-s + (−0.393 − 0.212i)10-s + 0.809i·11-s + (−0.0706 + 0.0464i)12-s + 0.277i·13-s + (−0.0262 + 0.0486i)14-s − 0.0377·15-s + (−0.396 + 0.917i)16-s + 0.242·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0864 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0864 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(104\)    =    \(2^{3} \cdot 13\)
Sign: $0.0864 + 0.996i$
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: \((2,\ 104,\ (\ :1/2),\ 0.0864 + 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.897996 - 0.823410i\)
\(L(\frac12)\) \(\approx\) \(0.897996 - 0.823410i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.671 + 1.24i)T \)
13 \( 1 - iT \)
good3 \( 1 + 0.146iT - 3T^{2} \)
5 \( 1 + iT - 5T^{2} \)
7 \( 1 + 0.146T + 7T^{2} \)
11 \( 1 - 2.68iT - 11T^{2} \)
17 \( 1 - T + 17T^{2} \)
19 \( 1 - 4iT - 19T^{2} \)
23 \( 1 + 6.68T + 23T^{2} \)
29 \( 1 + 4.39iT - 29T^{2} \)
31 \( 1 - 1.31T + 31T^{2} \)
37 \( 1 - 3.97iT - 37T^{2} \)
41 \( 1 + 6.39T + 41T^{2} \)
43 \( 1 + 6.83iT - 43T^{2} \)
47 \( 1 + 7.12T + 47T^{2} \)
53 \( 1 + 8.97iT - 53T^{2} \)
59 \( 1 + 12.3iT - 59T^{2} \)
61 \( 1 - 8.35iT - 61T^{2} \)
67 \( 1 - 8.29iT - 67T^{2} \)
71 \( 1 - 5.51T + 71T^{2} \)
73 \( 1 + 6.97T + 73T^{2} \)
79 \( 1 - 15.0T + 79T^{2} \)
83 \( 1 - 4.29iT - 83T^{2} \)
89 \( 1 + 5.37T + 89T^{2} \)
97 \( 1 + 10.3T + 97T^{2} \)
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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

−13.28789120651975842614937035296, −12.46783508721565093758017559821, −11.75031641771829656877295900432, −10.21736016246490816343404959248, −9.699328038323188754547216326617, −8.190602121309344737467922991517, −6.55795254465814190316149599944, −5.00314953812495710418064680919, −3.88839714389855242966581040494, −1.79574018365865677575886225674, 3.29149725194506718756651581044, 4.73811596476252295199356438168, 6.16958368614457738072186670479, 7.19247232960496672643276328721, 8.329944060218021981239546616889, 9.613372101299161699429239014357, 10.89926515501363429859278040410, 12.25012152703368611917126336259, 13.21464900121797240465517448943, 14.10393310103376478400064314210

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