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Properties

Label 2-1520-95.94-c0-0-3
Degree $2$
Conductor $1520$
Sign $1$
Analytic cond. $0.758578$
Root an. cond. $0.870964$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.41·3-s − 5-s + 1.00·9-s + 1.41·13-s − 1.41·15-s + 19-s + 25-s − 1.41·37-s + 2.00·39-s − 1.00·45-s + 49-s − 1.41·53-s + 1.41·57-s − 1.41·65-s − 1.41·67-s + 1.41·75-s − 0.999·81-s − 95-s − 1.41·97-s − 1.41·103-s − 1.41·107-s − 2.00·111-s + 1.41·113-s + 1.41·117-s + ⋯
L(s)  = 1  + 1.41·3-s − 5-s + 1.00·9-s + 1.41·13-s − 1.41·15-s + 19-s + 25-s − 1.41·37-s + 2.00·39-s − 1.00·45-s + 49-s − 1.41·53-s + 1.41·57-s − 1.41·65-s − 1.41·67-s + 1.41·75-s − 0.999·81-s − 95-s − 1.41·97-s − 1.41·103-s − 1.41·107-s − 2.00·111-s + 1.41·113-s + 1.41·117-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1520\)    =    \(2^{4} \cdot 5 \cdot 19\)
Sign: $1$
Analytic conductor: \(0.758578\)
Root analytic conductor: \(0.870964\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1520} (1329, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1520,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.567611674\)
\(L(\frac12)\) \(\approx\) \(1.567611674\)
\(L(1)\) 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 \( 1 + T \)
19 \( 1 - T \)
good3 \( 1 - 1.41T + T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 - 1.41T + T^{2} \)
17 \( 1 - T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + 1.41T + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + 1.41T + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + 1.41T + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + 1.41T + T^{2} \)
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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

−9.333758392986555810566549022775, −8.768366917604205082708066046311, −8.142098538207784704121976491787, −7.53483326984986013544304140303, −6.69205321015119439143585820856, −5.48376300487681058471845699583, −4.25691348345654187211046544339, −3.53075132896040688641337643238, −2.92013503580586984101221815322, −1.47050220859465655908048266546, 1.47050220859465655908048266546, 2.92013503580586984101221815322, 3.53075132896040688641337643238, 4.25691348345654187211046544339, 5.48376300487681058471845699583, 6.69205321015119439143585820856, 7.53483326984986013544304140303, 8.142098538207784704121976491787, 8.768366917604205082708066046311, 9.333758392986555810566549022775

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