Properties

Label 2-2415-1.1-c1-0-42
Degree $2$
Conductor $2415$
Sign $1$
Analytic cond. $19.2838$
Root an. cond. $4.39134$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.663·2-s + 3-s − 1.55·4-s + 5-s + 0.663·6-s − 7-s − 2.36·8-s + 9-s + 0.663·10-s + 5.33·11-s − 1.55·12-s + 1.57·13-s − 0.663·14-s + 15-s + 1.55·16-s − 1.91·17-s + 0.663·18-s − 0.933·19-s − 1.55·20-s − 21-s + 3.54·22-s + 23-s − 2.36·24-s + 25-s + 1.04·26-s + 27-s + 1.55·28-s + ⋯
L(s)  = 1  + 0.469·2-s + 0.577·3-s − 0.779·4-s + 0.447·5-s + 0.270·6-s − 0.377·7-s − 0.835·8-s + 0.333·9-s + 0.209·10-s + 1.60·11-s − 0.450·12-s + 0.436·13-s − 0.177·14-s + 0.258·15-s + 0.387·16-s − 0.464·17-s + 0.156·18-s − 0.214·19-s − 0.348·20-s − 0.218·21-s + 0.754·22-s + 0.208·23-s − 0.482·24-s + 0.200·25-s + 0.204·26-s + 0.192·27-s + 0.294·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2415\)    =    \(3 \cdot 5 \cdot 7 \cdot 23\)
Sign: $1$
Analytic conductor: \(19.2838\)
Root analytic conductor: \(4.39134\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2415,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.595743811\)
\(L(\frac12)\) \(\approx\) \(2.595743811\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
5 \( 1 - T \)
7 \( 1 + T \)
23 \( 1 - T \)
good2 \( 1 - 0.663T + 2T^{2} \)
11 \( 1 - 5.33T + 11T^{2} \)
13 \( 1 - 1.57T + 13T^{2} \)
17 \( 1 + 1.91T + 17T^{2} \)
19 \( 1 + 0.933T + 19T^{2} \)
29 \( 1 + 5.03T + 29T^{2} \)
31 \( 1 - 4.42T + 31T^{2} \)
37 \( 1 + 4.79T + 37T^{2} \)
41 \( 1 - 9.80T + 41T^{2} \)
43 \( 1 - 2.22T + 43T^{2} \)
47 \( 1 - 1.31T + 47T^{2} \)
53 \( 1 - 5.02T + 53T^{2} \)
59 \( 1 - 9.98T + 59T^{2} \)
61 \( 1 + 4.57T + 61T^{2} \)
67 \( 1 + 7.68T + 67T^{2} \)
71 \( 1 - 14.7T + 71T^{2} \)
73 \( 1 + 4.73T + 73T^{2} \)
79 \( 1 - 16.3T + 79T^{2} \)
83 \( 1 + 9.93T + 83T^{2} \)
89 \( 1 - 14.6T + 89T^{2} \)
97 \( 1 - 10.8T + 97T^{2} \)
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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

−9.038397708847487731255343676239, −8.491710807820412479649488732176, −7.36914497657082865620387315753, −6.44314201925834868743133138301, −5.92434822899115451487529283803, −4.83471885027087525595134902828, −3.98490050486332398591249604970, −3.48741999011067027433618927959, −2.28793953871104897860085475514, −0.989583567015228106429293292623, 0.989583567015228106429293292623, 2.28793953871104897860085475514, 3.48741999011067027433618927959, 3.98490050486332398591249604970, 4.83471885027087525595134902828, 5.92434822899115451487529283803, 6.44314201925834868743133138301, 7.36914497657082865620387315753, 8.491710807820412479649488732176, 9.038397708847487731255343676239

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