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Properties

Label 2-920-115.67-c1-0-11
Degree $2$
Conductor $920$
Sign $0.848 + 0.528i$
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  + (−2.26 − 1.69i)3-s + (2.20 − 0.358i)5-s + (0.137 + 1.92i)7-s + (1.40 + 4.78i)9-s + (−2.53 − 3.95i)11-s + (4.62 + 0.330i)13-s + (−5.60 − 2.92i)15-s + (1.23 + 3.32i)17-s + (3.07 + 6.73i)19-s + (2.94 − 4.58i)21-s + (2.02 + 4.34i)23-s + (4.74 − 1.58i)25-s + (1.96 − 5.27i)27-s + (−4.72 − 2.15i)29-s + (0.224 − 1.56i)31-s + ⋯
L(s)  = 1  + (−1.30 − 0.977i)3-s + (0.987 − 0.160i)5-s + (0.0520 + 0.727i)7-s + (0.468 + 1.59i)9-s + (−0.765 − 1.19i)11-s + (1.28 + 0.0917i)13-s + (−1.44 − 0.756i)15-s + (0.300 + 0.805i)17-s + (0.705 + 1.54i)19-s + (0.643 − 1.00i)21-s + (0.422 + 0.906i)23-s + (0.948 − 0.316i)25-s + (0.378 − 1.01i)27-s + (−0.876 − 0.400i)29-s + (0.0403 − 0.280i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.848 + 0.528i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.848 + 0.528i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(920\)    =    \(2^{3} \cdot 5 \cdot 23\)
Sign: $0.848 + 0.528i$
Analytic conductor: \(7.34623\)
Root analytic conductor: \(2.71039\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{920} (297, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 920,\ (\ :1/2),\ 0.848 + 0.528i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.19667 - 0.342141i\)
\(L(\frac12)\) \(\approx\) \(1.19667 - 0.342141i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.20 + 0.358i)T \)
23 \( 1 + (-2.02 - 4.34i)T \)
good3 \( 1 + (2.26 + 1.69i)T + (0.845 + 2.87i)T^{2} \)
7 \( 1 + (-0.137 - 1.92i)T + (-6.92 + 0.996i)T^{2} \)
11 \( 1 + (2.53 + 3.95i)T + (-4.56 + 10.0i)T^{2} \)
13 \( 1 + (-4.62 - 0.330i)T + (12.8 + 1.85i)T^{2} \)
17 \( 1 + (-1.23 - 3.32i)T + (-12.8 + 11.1i)T^{2} \)
19 \( 1 + (-3.07 - 6.73i)T + (-12.4 + 14.3i)T^{2} \)
29 \( 1 + (4.72 + 2.15i)T + (18.9 + 21.9i)T^{2} \)
31 \( 1 + (-0.224 + 1.56i)T + (-29.7 - 8.73i)T^{2} \)
37 \( 1 + (-1.83 + 3.36i)T + (-20.0 - 31.1i)T^{2} \)
41 \( 1 + (4.89 + 1.43i)T + (34.4 + 22.1i)T^{2} \)
43 \( 1 + (-7.30 + 9.76i)T + (-12.1 - 41.2i)T^{2} \)
47 \( 1 + (-8.52 - 8.52i)T + 47iT^{2} \)
53 \( 1 + (6.22 - 0.445i)T + (52.4 - 7.54i)T^{2} \)
59 \( 1 + (-6.53 + 5.66i)T + (8.39 - 58.3i)T^{2} \)
61 \( 1 + (-7.32 - 1.05i)T + (58.5 + 17.1i)T^{2} \)
67 \( 1 + (-7.24 + 1.57i)T + (60.9 - 27.8i)T^{2} \)
71 \( 1 + (5.94 + 3.82i)T + (29.4 + 64.5i)T^{2} \)
73 \( 1 + (-4.22 - 1.57i)T + (55.1 + 47.8i)T^{2} \)
79 \( 1 + (10.5 + 12.2i)T + (-11.2 + 78.1i)T^{2} \)
83 \( 1 + (5.33 + 2.91i)T + (44.8 + 69.8i)T^{2} \)
89 \( 1 + (0.876 + 6.09i)T + (-85.3 + 25.0i)T^{2} \)
97 \( 1 + (3.19 - 1.74i)T + (52.4 - 81.6i)T^{2} \)
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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

−10.27860616727030212391134051333, −9.108979109659613661101409418865, −8.256753263146022012694400038415, −7.37976101302461121933699793997, −6.04240381729199407598313397708, −5.83725563359732432995346809583, −5.41394999159305304436173601290, −3.57583728591537304060232638978, −2.02516657649952244243994631912, −1.04160053138585595681012196144, 0.951882798515722959334756066559, 2.76520960746279785106678619335, 4.19038410305076568968087473107, 5.02087516019120385124247904443, 5.56318865423928385045487526379, 6.66848104954973480990798943559, 7.24681787652302129961774978395, 8.795012622183599216089798066396, 9.745158287430514746983969782462, 10.12036775034760279134750557691

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