Properties

Label 2-7440-1.1-c1-0-5
Degree $2$
Conductor $7440$
Sign $1$
Analytic cond. $59.4086$
Root an. cond. $7.70770$
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  − 3-s + 5-s − 3.37·7-s + 9-s − 0.627·11-s − 2·13-s − 15-s − 4.74·17-s + 0.627·19-s + 3.37·21-s − 3.37·23-s + 25-s − 27-s − 8.74·29-s + 31-s + 0.627·33-s − 3.37·35-s − 0.744·37-s + 2·39-s + 0.744·41-s − 0.627·43-s + 45-s + 6.74·47-s + 4.37·49-s + 4.74·51-s + 1.37·53-s − 0.627·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s − 1.27·7-s + 0.333·9-s − 0.189·11-s − 0.554·13-s − 0.258·15-s − 1.15·17-s + 0.144·19-s + 0.735·21-s − 0.703·23-s + 0.200·25-s − 0.192·27-s − 1.62·29-s + 0.179·31-s + 0.109·33-s − 0.570·35-s − 0.122·37-s + 0.320·39-s + 0.116·41-s − 0.0957·43-s + 0.149·45-s + 0.983·47-s + 0.624·49-s + 0.664·51-s + 0.188·53-s − 0.0846·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7440\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 31\)
Sign: $1$
Analytic conductor: \(59.4086\)
Root analytic conductor: \(7.70770\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 7440,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7418154829\)
\(L(\frac12)\) \(\approx\) \(0.7418154829\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T \)
5 \( 1 - T \)
31 \( 1 - T \)
good7 \( 1 + 3.37T + 7T^{2} \)
11 \( 1 + 0.627T + 11T^{2} \)
13 \( 1 + 2T + 13T^{2} \)
17 \( 1 + 4.74T + 17T^{2} \)
19 \( 1 - 0.627T + 19T^{2} \)
23 \( 1 + 3.37T + 23T^{2} \)
29 \( 1 + 8.74T + 29T^{2} \)
37 \( 1 + 0.744T + 37T^{2} \)
41 \( 1 - 0.744T + 41T^{2} \)
43 \( 1 + 0.627T + 43T^{2} \)
47 \( 1 - 6.74T + 47T^{2} \)
53 \( 1 - 1.37T + 53T^{2} \)
59 \( 1 + 2.74T + 59T^{2} \)
61 \( 1 - 11.4T + 61T^{2} \)
67 \( 1 + 10.7T + 67T^{2} \)
71 \( 1 + 3.37T + 71T^{2} \)
73 \( 1 + 8.11T + 73T^{2} \)
79 \( 1 - 4.62T + 79T^{2} \)
83 \( 1 - 12T + 83T^{2} \)
89 \( 1 + 1.37T + 89T^{2} \)
97 \( 1 - 2T + 97T^{2} \)
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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

−7.66783905949348336052332401885, −7.10456628294832911001876840562, −6.40524094479749913293433631863, −5.91031111629671526535015958420, −5.21632208427655927573190629761, −4.34028292660842255750717023413, −3.59110078042870576215247492418, −2.63040779039133934182153000648, −1.86643750655640715952638526395, −0.42322978494496860810746944405, 0.42322978494496860810746944405, 1.86643750655640715952638526395, 2.63040779039133934182153000648, 3.59110078042870576215247492418, 4.34028292660842255750717023413, 5.21632208427655927573190629761, 5.91031111629671526535015958420, 6.40524094479749913293433631863, 7.10456628294832911001876840562, 7.66783905949348336052332401885

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