Properties

Label 2-624-12.11-c1-0-5
Degree $2$
Conductor $624$
Sign $0.866 - 0.5i$
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  + (−1.5 − 0.866i)3-s + 1.73i·5-s − 1.73i·7-s + (1.5 + 2.59i)9-s − 13-s + (1.49 − 2.59i)15-s + 1.73i·17-s + 3.46i·19-s + (−1.49 + 2.59i)21-s + 6·23-s + 2.00·25-s − 5.19i·27-s + 6.92i·29-s + 3.46i·31-s + 2.99·35-s + ⋯
L(s)  = 1  + (−0.866 − 0.499i)3-s + 0.774i·5-s − 0.654i·7-s + (0.5 + 0.866i)9-s − 0.277·13-s + (0.387 − 0.670i)15-s + 0.420i·17-s + 0.794i·19-s + (−0.327 + 0.566i)21-s + 1.25·23-s + 0.400·25-s − 0.999i·27-s + 1.28i·29-s + 0.622i·31-s + 0.507·35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 - 0.5i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.866 - 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $0.866 - 0.5i$
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: \((2,\ 624,\ (\ :1/2),\ 0.866 - 0.5i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.01958 + 0.273196i\)
\(L(\frac12)\) \(\approx\) \(1.01958 + 0.273196i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.5 + 0.866i)T \)
13 \( 1 + T \)
good5 \( 1 - 1.73iT - 5T^{2} \)
7 \( 1 + 1.73iT - 7T^{2} \)
11 \( 1 + 11T^{2} \)
17 \( 1 - 1.73iT - 17T^{2} \)
19 \( 1 - 3.46iT - 19T^{2} \)
23 \( 1 - 6T + 23T^{2} \)
29 \( 1 - 6.92iT - 29T^{2} \)
31 \( 1 - 3.46iT - 31T^{2} \)
37 \( 1 - 11T + 37T^{2} \)
41 \( 1 + 3.46iT - 41T^{2} \)
43 \( 1 - 1.73iT - 43T^{2} \)
47 \( 1 - 9T + 47T^{2} \)
53 \( 1 - 3.46iT - 53T^{2} \)
59 \( 1 + 12T + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 - 10.3iT - 67T^{2} \)
71 \( 1 + 3T + 71T^{2} \)
73 \( 1 - 4T + 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + 10.3iT - 89T^{2} \)
97 \( 1 + 8T + 97T^{2} \)
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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

−10.69278005903977297106871428806, −10.23678175086013798097641279074, −8.949049855542413340453233797162, −7.67086701624874893266228282852, −7.10084586049746251980927493452, −6.31280552217635835200433696654, −5.30796703796158874061048958489, −4.19181892945315703687364056662, −2.82949436860675664543436214061, −1.22806379712622037220416208836, 0.77891088897599007283079227152, 2.69156371615424922345819539367, 4.28077129458403140817299104907, 5.00009315640912756947888693595, 5.81720253103164057555697317854, 6.79600227196598273710751089560, 7.947427797170295065900704670126, 9.246146081989699745371023490426, 9.368802879072057133690714691291, 10.66507103480687716051500418936

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