Properties

Label 2-920-5.4-c1-0-21
Degree $2$
Conductor $920$
Sign $0.263 + 0.964i$
Analytic cond. $7.34623$
Root an. cond. $2.71039$
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  − 1.69i·3-s + (0.589 + 2.15i)5-s − 4.22i·7-s + 0.118·9-s + 4.59·11-s − 0.978i·13-s + (3.66 − 1.00i)15-s + 3.04i·17-s − 1.91·19-s − 7.17·21-s + i·23-s + (−4.30 + 2.54i)25-s − 5.29i·27-s + 0.737·29-s + 2.97·31-s + ⋯
L(s)  = 1  − 0.980i·3-s + (0.263 + 0.964i)5-s − 1.59i·7-s + 0.0394·9-s + 1.38·11-s − 0.271i·13-s + (0.945 − 0.258i)15-s + 0.737i·17-s − 0.439·19-s − 1.56·21-s + 0.208i·23-s + (−0.861 + 0.508i)25-s − 1.01i·27-s + 0.136·29-s + 0.535·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.263 + 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.263 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(920\)    =    \(2^{3} \cdot 5 \cdot 23\)
Sign: $0.263 + 0.964i$
Analytic conductor: \(7.34623\)
Root analytic conductor: \(2.71039\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{920} (369, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 920,\ (\ :1/2),\ 0.263 + 0.964i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.42678 - 1.08920i\)
\(L(\frac12)\) \(\approx\) \(1.42678 - 1.08920i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-0.589 - 2.15i)T \)
23 \( 1 - iT \)
good3 \( 1 + 1.69iT - 3T^{2} \)
7 \( 1 + 4.22iT - 7T^{2} \)
11 \( 1 - 4.59T + 11T^{2} \)
13 \( 1 + 0.978iT - 13T^{2} \)
17 \( 1 - 3.04iT - 17T^{2} \)
19 \( 1 + 1.91T + 19T^{2} \)
29 \( 1 - 0.737T + 29T^{2} \)
31 \( 1 - 2.97T + 31T^{2} \)
37 \( 1 + 8.93iT - 37T^{2} \)
41 \( 1 - 9.08T + 41T^{2} \)
43 \( 1 + 6.97iT - 43T^{2} \)
47 \( 1 + 2.58iT - 47T^{2} \)
53 \( 1 + 2.71iT - 53T^{2} \)
59 \( 1 + 7.13T + 59T^{2} \)
61 \( 1 + 0.731T + 61T^{2} \)
67 \( 1 - 7.16iT - 67T^{2} \)
71 \( 1 + 6.08T + 71T^{2} \)
73 \( 1 + 5.96iT - 73T^{2} \)
79 \( 1 - 2.06T + 79T^{2} \)
83 \( 1 - 4.39iT - 83T^{2} \)
89 \( 1 - 7.25T + 89T^{2} \)
97 \( 1 - 7.04iT - 97T^{2} \)
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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

−10.14063622148230619312332195144, −9.105273641273021023785693665016, −7.84593323128568022909545978056, −7.26504814960319616038212427417, −6.64371947934869518465693872976, −6.01310022884702636271901755491, −4.24569909867365501983017787992, −3.61567465686504455629370668796, −2.07164642597222462659722304047, −0.973388223385464168789223950921, 1.51508009205966393609671318970, 2.88896906212880176879135771713, 4.27865648064786237548810794253, 4.80187812267479306740600398323, 5.80955592738332930276437125295, 6.56681537197885211116036007627, 8.064314668208737394803456869952, 9.019837344541166781318177567657, 9.246485869055617753570686231731, 9.912240985702542231350047870665

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