Properties

Label 2-2415-1.1-c1-0-72
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  + 2.42·2-s + 3-s + 3.86·4-s + 5-s + 2.42·6-s − 7-s + 4.50·8-s + 9-s + 2.42·10-s + 2.79·11-s + 3.86·12-s − 5.96·13-s − 2.42·14-s + 15-s + 3.18·16-s + 4.49·17-s + 2.42·18-s + 8.29·19-s + 3.86·20-s − 21-s + 6.76·22-s + 23-s + 4.50·24-s + 25-s − 14.4·26-s + 27-s − 3.86·28-s + ⋯
L(s)  = 1  + 1.71·2-s + 0.577·3-s + 1.93·4-s + 0.447·5-s + 0.988·6-s − 0.377·7-s + 1.59·8-s + 0.333·9-s + 0.765·10-s + 0.842·11-s + 1.11·12-s − 1.65·13-s − 0.646·14-s + 0.258·15-s + 0.795·16-s + 1.09·17-s + 0.570·18-s + 1.90·19-s + 0.863·20-s − 0.218·21-s + 1.44·22-s + 0.208·23-s + 0.919·24-s + 0.200·25-s − 2.83·26-s + 0.192·27-s − 0.729·28-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

\(L(1)\) \(\approx\) \(6.655444875\)
\(L(\frac12)\) \(\approx\) \(6.655444875\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.42T + 2T^{2} \)
11 \( 1 - 2.79T + 11T^{2} \)
13 \( 1 + 5.96T + 13T^{2} \)
17 \( 1 - 4.49T + 17T^{2} \)
19 \( 1 - 8.29T + 19T^{2} \)
29 \( 1 + 0.288T + 29T^{2} \)
31 \( 1 - 6.34T + 31T^{2} \)
37 \( 1 + 11.0T + 37T^{2} \)
41 \( 1 + 8.10T + 41T^{2} \)
43 \( 1 + 0.264T + 43T^{2} \)
47 \( 1 - 3.28T + 47T^{2} \)
53 \( 1 + 4.34T + 53T^{2} \)
59 \( 1 - 15.0T + 59T^{2} \)
61 \( 1 + 7.31T + 61T^{2} \)
67 \( 1 - 2.53T + 67T^{2} \)
71 \( 1 - 10.2T + 71T^{2} \)
73 \( 1 + 4.95T + 73T^{2} \)
79 \( 1 + 16.7T + 79T^{2} \)
83 \( 1 - 3.16T + 83T^{2} \)
89 \( 1 - 3.44T + 89T^{2} \)
97 \( 1 + 8.27T + 97T^{2} \)
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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

−9.113969452578834847651896205547, −7.891197094279031102259257274422, −7.05226656463285299835135525097, −6.65680452194706990355515025696, −5.38180290092489709168524328359, −5.19194447539232595761022585808, −4.05723701831019681276497910727, −3.22641134148286708355922430123, −2.68701059242915819186467737345, −1.48611365306360526968184044986, 1.48611365306360526968184044986, 2.68701059242915819186467737345, 3.22641134148286708355922430123, 4.05723701831019681276497910727, 5.19194447539232595761022585808, 5.38180290092489709168524328359, 6.65680452194706990355515025696, 7.05226656463285299835135525097, 7.891197094279031102259257274422, 9.113969452578834847651896205547

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