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Properties

Label 2-920-115.102-c1-0-26
Degree $2$
Conductor $920$
Sign $0.640 + 0.768i$
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  + (2.10 − 0.458i)3-s + (−2.23 − 0.0896i)5-s + (1.66 − 0.908i)7-s + (1.49 − 0.684i)9-s + (−0.566 − 0.490i)11-s + (1.53 − 2.80i)13-s + (−4.74 + 0.835i)15-s + (5.94 + 4.45i)17-s + (0.974 − 6.77i)19-s + (3.08 − 2.67i)21-s + (0.956 − 4.69i)23-s + (4.98 + 0.400i)25-s + (−2.33 + 1.74i)27-s + (−4.82 + 0.693i)29-s + (7.40 − 4.75i)31-s + ⋯
L(s)  = 1  + (1.21 − 0.264i)3-s + (−0.999 − 0.0400i)5-s + (0.628 − 0.343i)7-s + (0.499 − 0.228i)9-s + (−0.170 − 0.147i)11-s + (0.424 − 0.778i)13-s + (−1.22 + 0.215i)15-s + (1.44 + 1.08i)17-s + (0.223 − 1.55i)19-s + (0.674 − 0.584i)21-s + (0.199 − 0.979i)23-s + (0.996 + 0.0800i)25-s + (−0.448 + 0.336i)27-s + (−0.896 + 0.128i)29-s + (1.32 − 0.854i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.640 + 0.768i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.640 + 0.768i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(920\)    =    \(2^{3} \cdot 5 \cdot 23\)
Sign: $0.640 + 0.768i$
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: \((2,\ 920,\ (\ :1/2),\ 0.640 + 0.768i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.94541 - 0.911182i\)
\(L(\frac12)\) \(\approx\) \(1.94541 - 0.911182i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.23 + 0.0896i)T \)
23 \( 1 + (-0.956 + 4.69i)T \)
good3 \( 1 + (-2.10 + 0.458i)T + (2.72 - 1.24i)T^{2} \)
7 \( 1 + (-1.66 + 0.908i)T + (3.78 - 5.88i)T^{2} \)
11 \( 1 + (0.566 + 0.490i)T + (1.56 + 10.8i)T^{2} \)
13 \( 1 + (-1.53 + 2.80i)T + (-7.02 - 10.9i)T^{2} \)
17 \( 1 + (-5.94 - 4.45i)T + (4.78 + 16.3i)T^{2} \)
19 \( 1 + (-0.974 + 6.77i)T + (-18.2 - 5.35i)T^{2} \)
29 \( 1 + (4.82 - 0.693i)T + (27.8 - 8.17i)T^{2} \)
31 \( 1 + (-7.40 + 4.75i)T + (12.8 - 28.1i)T^{2} \)
37 \( 1 + (-0.138 - 0.0518i)T + (27.9 + 24.2i)T^{2} \)
41 \( 1 + (-1.76 + 3.85i)T + (-26.8 - 30.9i)T^{2} \)
43 \( 1 + (-0.921 - 4.23i)T + (-39.1 + 17.8i)T^{2} \)
47 \( 1 + (2.27 + 2.27i)T + 47iT^{2} \)
53 \( 1 + (1.41 + 2.59i)T + (-28.6 + 44.5i)T^{2} \)
59 \( 1 + (-1.51 - 5.14i)T + (-49.6 + 31.8i)T^{2} \)
61 \( 1 + (5.46 + 8.50i)T + (-25.3 + 55.4i)T^{2} \)
67 \( 1 + (0.820 + 0.0587i)T + (66.3 + 9.53i)T^{2} \)
71 \( 1 + (-9.72 - 11.2i)T + (-10.1 + 70.2i)T^{2} \)
73 \( 1 + (-7.62 - 10.1i)T + (-20.5 + 70.0i)T^{2} \)
79 \( 1 + (10.0 - 2.94i)T + (66.4 - 42.7i)T^{2} \)
83 \( 1 + (2.56 - 6.88i)T + (-62.7 - 54.3i)T^{2} \)
89 \( 1 + (11.9 + 7.64i)T + (36.9 + 80.9i)T^{2} \)
97 \( 1 + (2.23 + 5.98i)T + (-73.3 + 63.5i)T^{2} \)
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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

−9.885214895064861184146118864290, −8.810305890360650187104399215049, −8.075189587353577783417871650580, −7.87431324935231451495732409905, −6.87251119869479626244641006818, −5.51622941315252275159576925877, −4.37286155133148514402047617209, −3.45729245171214361842818474236, −2.62570933768487772976224327139, −1.00659893824100348348475965762, 1.57009395509988906240510410633, 3.03331016703905474334632155418, 3.65352022840631006897038651113, 4.68326779570017284923901997560, 5.75634384064089284280094033620, 7.20047955468075478864693107407, 7.916842893807118413051940672745, 8.342226491546230059956840051656, 9.312850859143177812988840823895, 9.928234960269233902089452652510

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