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Properties

Label 2-624-13.10-c1-0-1
Degree $2$
Conductor $624$
Sign $-0.903 - 0.427i$
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  + (0.5 + 0.866i)3-s + 4.14i·5-s + (−2.08 − 1.20i)7-s + (−0.499 + 0.866i)9-s + (−3 + 1.73i)11-s + (1.5 − 3.27i)13-s + (−3.58 + 2.07i)15-s + (−2.58 + 4.48i)17-s + (3 + 1.73i)19-s − 2.41i·21-s + (−1 − 1.73i)23-s − 12.1·25-s − 0.999·27-s + (−1.58 − 2.75i)29-s − 1.05i·31-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + 1.85i·5-s + (−0.789 − 0.455i)7-s + (−0.166 + 0.288i)9-s + (−0.904 + 0.522i)11-s + (0.416 − 0.909i)13-s + (−0.926 + 0.535i)15-s + (−0.628 + 1.08i)17-s + (0.688 + 0.397i)19-s − 0.526i·21-s + (−0.208 − 0.361i)23-s − 2.43·25-s − 0.192·27-s + (−0.295 − 0.511i)29-s − 0.188i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.903 - 0.427i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.903 - 0.427i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $-0.903 - 0.427i$
Analytic conductor: \(4.98266\)
Root analytic conductor: \(2.23218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{624} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 624,\ (\ :1/2),\ -0.903 - 0.427i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.228232 + 1.01611i\)
\(L(\frac12)\) \(\approx\) \(0.228232 + 1.01611i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (-1.5 + 3.27i)T \)
good5 \( 1 - 4.14iT - 5T^{2} \)
7 \( 1 + (2.08 + 1.20i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (3 - 1.73i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (2.58 - 4.48i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3 - 1.73i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1 + 1.73i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.58 + 2.75i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 1.05iT - 31T^{2} \)
37 \( 1 + (-6.58 + 3.80i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.589 - 0.340i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (6.08 - 10.5i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 10.3iT - 47T^{2} \)
53 \( 1 - 1.17T + 53T^{2} \)
59 \( 1 + (-10.1 - 5.87i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.5 - 4.33i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.08 + 1.20i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3 - 1.73i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 14.8iT - 73T^{2} \)
79 \( 1 + 1.82T + 79T^{2} \)
83 \( 1 - 1.36iT - 83T^{2} \)
89 \( 1 + (-6 + 3.46i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-15.2 - 8.81i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.73662451798685408467736902547, −10.16835017978187438239547528068, −9.682297312143627903574766753984, −8.132754391952868433076980210033, −7.50110727882819580970816129748, −6.49719072948723854943559932617, −5.76615713344046765367871430018, −4.13804182796208055081847248238, −3.25543990982643354458259183654, −2.46661183239235434790481378210, 0.52630323634850467010380576882, 2.04672257741591743456782058672, 3.45742419728621811006553504619, 4.83152319307036030528869144853, 5.52357018246725279757237603110, 6.64637501955459232843891063324, 7.74809990354575146000964565548, 8.729733745247696357358362606565, 9.079523657592813061219498361441, 9.906877463931391564388124450578

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