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Properties

Label 2-104-104.11-c1-0-11
Degree $2$
Conductor $104$
Sign $-0.326 + 0.945i$
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  + (0.869 − 1.11i)2-s + (−0.928 − 1.60i)3-s + (−0.488 − 1.93i)4-s + (−1.51 + 1.51i)5-s + (−2.60 − 0.362i)6-s + (2.97 − 0.797i)7-s + (−2.58 − 1.14i)8-s + (−0.223 + 0.386i)9-s + (0.373 + 3.01i)10-s + (0.865 + 0.231i)11-s + (−2.66 + 2.58i)12-s + (0.159 + 3.60i)13-s + (1.69 − 4.01i)14-s + (3.84 + 1.03i)15-s + (−3.52 + 1.89i)16-s + (1.05 + 0.610i)17-s + ⋯
L(s)  = 1  + (0.614 − 0.788i)2-s + (−0.535 − 0.928i)3-s + (−0.244 − 0.969i)4-s + (−0.678 + 0.678i)5-s + (−1.06 − 0.147i)6-s + (1.12 − 0.301i)7-s + (−0.915 − 0.403i)8-s + (−0.0743 + 0.128i)9-s + (0.118 + 0.952i)10-s + (0.260 + 0.0699i)11-s + (−0.769 + 0.746i)12-s + (0.0442 + 0.999i)13-s + (0.453 − 1.07i)14-s + (0.994 + 0.266i)15-s + (−0.880 + 0.473i)16-s + (0.256 + 0.147i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.326 + 0.945i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.326 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(104\)    =    \(2^{3} \cdot 13\)
Sign: $-0.326 + 0.945i$
Analytic conductor: \(0.830444\)
Root analytic conductor: \(0.911287\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{104} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 104,\ (\ :1/2),\ -0.326 + 0.945i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.657222 - 0.922141i\)
\(L(\frac12)\) \(\approx\) \(0.657222 - 0.922141i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.869 + 1.11i)T \)
13 \( 1 + (-0.159 - 3.60i)T \)
good3 \( 1 + (0.928 + 1.60i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (1.51 - 1.51i)T - 5iT^{2} \)
7 \( 1 + (-2.97 + 0.797i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (-0.865 - 0.231i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-1.05 - 0.610i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-6.68 + 1.79i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (0.433 + 0.751i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.26 - 1.88i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (5.06 - 5.06i)T - 31iT^{2} \)
37 \( 1 + (-2.52 + 9.43i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (3.12 - 11.6i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (4.22 + 2.44i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.24 + 4.24i)T + 47iT^{2} \)
53 \( 1 - 2.16iT - 53T^{2} \)
59 \( 1 + (-0.0382 - 0.142i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-7.26 - 4.19i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.422 + 1.57i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (4.27 + 15.9i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-9.55 + 9.55i)T - 73iT^{2} \)
79 \( 1 - 6.37iT - 79T^{2} \)
83 \( 1 + (-3.53 - 3.53i)T + 83iT^{2} \)
89 \( 1 + (-9.33 - 2.50i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (10.6 - 2.86i)T + (84.0 - 48.5i)T^{2} \)
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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

−13.38126874980854370540114489243, −12.14774948051026850515023922911, −11.51492815850376479340702276192, −10.93886606781346774119127197702, −9.358034833220400593297903438240, −7.58627863097749017243180516282, −6.67913213558602861794113695080, −5.15272395106592533867287135385, −3.66553322783643420189249603878, −1.55331349765993160380799661205, 3.77223214592484559181249223231, 4.96132286343488342166732347521, 5.56148797802440274416639100633, 7.61226030317507604084485027637, 8.340318368047579157535763433875, 9.749906206646843760407050778881, 11.33994929119832878447308859766, 11.89822782025593252266833616997, 13.13368608891038586637099026591, 14.38171155400553064753297036999

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