Properties

Label 2-2415-1.1-c1-0-26
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  − 0.866·2-s + 3-s − 1.24·4-s + 5-s − 0.866·6-s − 7-s + 2.81·8-s + 9-s − 0.866·10-s + 3.46·11-s − 1.24·12-s − 5.43·13-s + 0.866·14-s + 15-s + 0.0595·16-s + 7.18·17-s − 0.866·18-s − 6.78·19-s − 1.24·20-s − 21-s − 3.00·22-s + 23-s + 2.81·24-s + 25-s + 4.70·26-s + 27-s + 1.24·28-s + ⋯
L(s)  = 1  − 0.612·2-s + 0.577·3-s − 0.624·4-s + 0.447·5-s − 0.353·6-s − 0.377·7-s + 0.995·8-s + 0.333·9-s − 0.273·10-s + 1.04·11-s − 0.360·12-s − 1.50·13-s + 0.231·14-s + 0.258·15-s + 0.0148·16-s + 1.74·17-s − 0.204·18-s − 1.55·19-s − 0.279·20-s − 0.218·21-s − 0.640·22-s + 0.208·23-s + 0.574·24-s + 0.200·25-s + 0.923·26-s + 0.192·27-s + 0.236·28-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

\(L(1)\) \(\approx\) \(1.427693492\)
\(L(\frac12)\) \(\approx\) \(1.427693492\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.866T + 2T^{2} \)
11 \( 1 - 3.46T + 11T^{2} \)
13 \( 1 + 5.43T + 13T^{2} \)
17 \( 1 - 7.18T + 17T^{2} \)
19 \( 1 + 6.78T + 19T^{2} \)
29 \( 1 - 7.97T + 29T^{2} \)
31 \( 1 + 4.50T + 31T^{2} \)
37 \( 1 - 4.27T + 37T^{2} \)
41 \( 1 - 11.4T + 41T^{2} \)
43 \( 1 - 2.81T + 43T^{2} \)
47 \( 1 + 5.16T + 47T^{2} \)
53 \( 1 + 12.7T + 53T^{2} \)
59 \( 1 - 5.33T + 59T^{2} \)
61 \( 1 - 3.88T + 61T^{2} \)
67 \( 1 - 8.15T + 67T^{2} \)
71 \( 1 + 14.2T + 71T^{2} \)
73 \( 1 - 10.4T + 73T^{2} \)
79 \( 1 + 10.3T + 79T^{2} \)
83 \( 1 - 4.29T + 83T^{2} \)
89 \( 1 - 17.5T + 89T^{2} \)
97 \( 1 - 0.254T + 97T^{2} \)
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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

−9.141647223802420814445299204790, −8.263126284238467235345608984592, −7.64575043145742763661871458549, −6.82328786863987414870399437762, −5.91602887556424061313020462513, −4.84327632496079977235662543023, −4.16247626195721084757772270497, −3.11815134469545884886863498508, −2.02630723029532199063809469473, −0.842692328272881131353268954592, 0.842692328272881131353268954592, 2.02630723029532199063809469473, 3.11815134469545884886863498508, 4.16247626195721084757772270497, 4.84327632496079977235662543023, 5.91602887556424061313020462513, 6.82328786863987414870399437762, 7.64575043145742763661871458549, 8.263126284238467235345608984592, 9.141647223802420814445299204790

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