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Properties

Label 2-624-12.11-c1-0-14
Degree $2$
Conductor $624$
Sign $1$
Analytic cond. $4.98266$
Root an. cond. $2.23218$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.73·3-s − 2.44i·5-s + 4.24i·7-s + 2.99·9-s + 3.46·11-s − 13-s − 4.24i·15-s + 4.89i·17-s − 4.24i·19-s + 7.34i·21-s + 3.46·23-s − 0.999·25-s + 5.19·27-s − 9.79i·29-s + 4.24i·31-s + ⋯
L(s)  = 1  + 1.00·3-s − 1.09i·5-s + 1.60i·7-s + 0.999·9-s + 1.04·11-s − 0.277·13-s − 1.09i·15-s + 1.18i·17-s − 0.973i·19-s + 1.60i·21-s + 0.722·23-s − 0.199·25-s + 1.00·27-s − 1.81i·29-s + 0.762i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $1$
Analytic conductor: \(4.98266\)
Root analytic conductor: \(2.23218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{624} (287, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 624,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.20255\)
\(L(\frac12)\) \(\approx\) \(2.20255\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.73T \)
13 \( 1 + T \)
good5 \( 1 + 2.44iT - 5T^{2} \)
7 \( 1 - 4.24iT - 7T^{2} \)
11 \( 1 - 3.46T + 11T^{2} \)
17 \( 1 - 4.89iT - 17T^{2} \)
19 \( 1 + 4.24iT - 19T^{2} \)
23 \( 1 - 3.46T + 23T^{2} \)
29 \( 1 + 9.79iT - 29T^{2} \)
31 \( 1 - 4.24iT - 31T^{2} \)
37 \( 1 + 10T + 37T^{2} \)
41 \( 1 + 2.44iT - 41T^{2} \)
43 \( 1 + 8.48iT - 43T^{2} \)
47 \( 1 - 3.46T + 47T^{2} \)
53 \( 1 - 9.79iT - 53T^{2} \)
59 \( 1 + 3.46T + 59T^{2} \)
61 \( 1 + 4T + 61T^{2} \)
67 \( 1 - 12.7iT - 67T^{2} \)
71 \( 1 + 10.3T + 71T^{2} \)
73 \( 1 + 2T + 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 + 3.46T + 83T^{2} \)
89 \( 1 + 7.34iT - 89T^{2} \)
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

−10.40550719683333931210955921431, −9.257327527428587604706289406772, −8.860372790305197529840925190152, −8.452745294976999189781289641058, −7.17260940484519703021314731042, −6.02650392045177679831856528708, −4.98724391273870609934861813565, −3.99916984815724606028175961725, −2.66687567682660406214788202417, −1.55494070915309958371517701366, 1.43337312677129174398634733095, 3.06868816581681902145613938632, 3.69156429925253801044711362234, 4.76857519598979506379846293900, 6.63185245835963511690929692209, 7.08985852658950484904150105806, 7.73901721894784803796882009218, 8.967177787367451520654075170187, 9.805015065839330745369492258422, 10.49735279463980769651514964979

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