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Properties

Label 2-920-115.102-c1-0-25
Degree $2$
Conductor $920$
Sign $0.274 + 0.961i$
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  + (−0.507 + 0.110i)3-s + (−0.763 + 2.10i)5-s + (3.36 − 1.83i)7-s + (−2.48 + 1.13i)9-s + (−4.65 − 4.03i)11-s + (1.62 − 2.96i)13-s + (0.155 − 1.15i)15-s + (−2.47 − 1.85i)17-s + (0.331 − 2.30i)19-s + (−1.50 + 1.30i)21-s + (4.56 + 1.45i)23-s + (−3.83 − 3.20i)25-s + (2.38 − 1.78i)27-s + (7.35 − 1.05i)29-s + (0.738 − 0.474i)31-s + ⋯
L(s)  = 1  + (−0.293 + 0.0637i)3-s + (−0.341 + 0.939i)5-s + (1.27 − 0.694i)7-s + (−0.827 + 0.377i)9-s + (−1.40 − 1.21i)11-s + (0.449 − 0.823i)13-s + (0.0401 − 0.297i)15-s + (−0.600 − 0.449i)17-s + (0.0760 − 0.528i)19-s + (−0.328 + 0.284i)21-s + (0.952 + 0.303i)23-s + (−0.766 − 0.641i)25-s + (0.458 − 0.343i)27-s + (1.36 − 0.196i)29-s + (0.132 − 0.0852i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.274 + 0.961i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.274 + 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(920\)    =    \(2^{3} \cdot 5 \cdot 23\)
Sign: $0.274 + 0.961i$
Analytic conductor: \(7.34623\)
Root analytic conductor: \(2.71039\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{920} (217, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 920,\ (\ :1/2),\ 0.274 + 0.961i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.829987 - 0.626035i\)
\(L(\frac12)\) \(\approx\) \(0.829987 - 0.626035i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.763 - 2.10i)T \)
23 \( 1 + (-4.56 - 1.45i)T \)
good3 \( 1 + (0.507 - 0.110i)T + (2.72 - 1.24i)T^{2} \)
7 \( 1 + (-3.36 + 1.83i)T + (3.78 - 5.88i)T^{2} \)
11 \( 1 + (4.65 + 4.03i)T + (1.56 + 10.8i)T^{2} \)
13 \( 1 + (-1.62 + 2.96i)T + (-7.02 - 10.9i)T^{2} \)
17 \( 1 + (2.47 + 1.85i)T + (4.78 + 16.3i)T^{2} \)
19 \( 1 + (-0.331 + 2.30i)T + (-18.2 - 5.35i)T^{2} \)
29 \( 1 + (-7.35 + 1.05i)T + (27.8 - 8.17i)T^{2} \)
31 \( 1 + (-0.738 + 0.474i)T + (12.8 - 28.1i)T^{2} \)
37 \( 1 + (3.59 + 1.34i)T + (27.9 + 24.2i)T^{2} \)
41 \( 1 + (0.404 - 0.885i)T + (-26.8 - 30.9i)T^{2} \)
43 \( 1 + (2.19 + 10.1i)T + (-39.1 + 17.8i)T^{2} \)
47 \( 1 + (-8.05 - 8.05i)T + 47iT^{2} \)
53 \( 1 + (1.86 + 3.41i)T + (-28.6 + 44.5i)T^{2} \)
59 \( 1 + (0.365 + 1.24i)T + (-49.6 + 31.8i)T^{2} \)
61 \( 1 + (7.02 + 10.9i)T + (-25.3 + 55.4i)T^{2} \)
67 \( 1 + (-0.964 - 0.0689i)T + (66.3 + 9.53i)T^{2} \)
71 \( 1 + (-0.390 - 0.450i)T + (-10.1 + 70.2i)T^{2} \)
73 \( 1 + (8.56 + 11.4i)T + (-20.5 + 70.0i)T^{2} \)
79 \( 1 + (-16.1 + 4.74i)T + (66.4 - 42.7i)T^{2} \)
83 \( 1 + (0.553 - 1.48i)T + (-62.7 - 54.3i)T^{2} \)
89 \( 1 + (2.95 + 1.89i)T + (36.9 + 80.9i)T^{2} \)
97 \( 1 + (-3.79 - 10.1i)T + (-73.3 + 63.5i)T^{2} \)
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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

−10.49421886549990733792624440256, −8.827176652828339630433037161428, −8.052328799718915427058257792107, −7.60493156721944124087393043880, −6.46339796881544328870986416051, −5.42293064627785363943394351053, −4.77873441329800036304143994750, −3.32575986805264033691546297577, −2.54772747387800182855397659262, −0.52545398109216637959170921575, 1.46253964839161130344403604685, 2.62613227975314911895485293026, 4.34391378659652032605676712682, 4.96883884960950890593905088290, 5.66452605246710728445252993519, 6.87788871733612327363584592289, 8.003482781110123920218394439961, 8.516234519430689215350429660383, 9.161308470272534719012295618164, 10.38627450346569168369468196996

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