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Properties

Label 2-624-208.187-c1-0-15
Degree $2$
Conductor $624$
Sign $-0.454 - 0.890i$
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  + (0.924 + 1.07i)2-s + (−0.707 − 0.707i)3-s + (−0.292 + 1.97i)4-s + 0.424·5-s + (0.103 − 1.41i)6-s + (2.87 + 2.87i)7-s + (−2.38 + 1.51i)8-s + 1.00i·9-s + (0.392 + 0.454i)10-s − 5.66·11-s + (1.60 − 1.19i)12-s + (3.59 − 0.284i)13-s + (−0.421 + 5.73i)14-s + (−0.300 − 0.300i)15-s + (−3.82 − 1.15i)16-s + 0.899i·17-s + ⋯
L(s)  = 1  + (0.653 + 0.757i)2-s + (−0.408 − 0.408i)3-s + (−0.146 + 0.989i)4-s + 0.189·5-s + (0.0423 − 0.575i)6-s + (1.08 + 1.08i)7-s + (−0.844 + 0.535i)8-s + 0.333i·9-s + (0.124 + 0.143i)10-s − 1.70·11-s + (0.463 − 0.344i)12-s + (0.996 − 0.0788i)13-s + (−0.112 + 1.53i)14-s + (−0.0774 − 0.0774i)15-s + (−0.957 − 0.289i)16-s + 0.218i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.454 - 0.890i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.454 - 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $-0.454 - 0.890i$
Analytic conductor: \(4.98266\)
Root analytic conductor: \(2.23218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{624} (187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 624,\ (\ :1/2),\ -0.454 - 0.890i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.931882 + 1.52234i\)
\(L(\frac12)\) \(\approx\) \(0.931882 + 1.52234i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.924 - 1.07i)T \)
3 \( 1 + (0.707 + 0.707i)T \)
13 \( 1 + (-3.59 + 0.284i)T \)
good5 \( 1 - 0.424T + 5T^{2} \)
7 \( 1 + (-2.87 - 2.87i)T + 7iT^{2} \)
11 \( 1 + 5.66T + 11T^{2} \)
17 \( 1 - 0.899iT - 17T^{2} \)
19 \( 1 - 3.25T + 19T^{2} \)
23 \( 1 - 8.85iT - 23T^{2} \)
29 \( 1 + (-2.63 + 2.63i)T - 29iT^{2} \)
31 \( 1 + (-3.46 - 3.46i)T + 31iT^{2} \)
37 \( 1 - 4.35iT - 37T^{2} \)
41 \( 1 + (3.94 + 3.94i)T + 41iT^{2} \)
43 \( 1 + (1.17 + 1.17i)T + 43iT^{2} \)
47 \( 1 + (0.575 - 0.575i)T - 47iT^{2} \)
53 \( 1 + (-2.99 - 2.99i)T + 53iT^{2} \)
59 \( 1 - 2.58T + 59T^{2} \)
61 \( 1 + (-9.73 + 9.73i)T - 61iT^{2} \)
67 \( 1 + 4.69T + 67T^{2} \)
71 \( 1 + (-0.830 + 0.830i)T - 71iT^{2} \)
73 \( 1 + (-2.26 + 2.26i)T - 73iT^{2} \)
79 \( 1 + 7.12iT - 79T^{2} \)
83 \( 1 - 3.55T + 83T^{2} \)
89 \( 1 + (-7.19 + 7.19i)T - 89iT^{2} \)
97 \( 1 + (-8.58 + 8.58i)T - 97iT^{2} \)
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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

−11.31095564059054392067358298652, −10.06086109804260618962916136481, −8.715433409518803966338535434060, −8.055903637528306523317620819607, −7.44841998388518251144254807446, −6.07793016927566223919763702788, −5.47237121322249193000410051000, −4.89546467074725554305615703128, −3.27585680858020642621550963274, −1.99080084344563392587252977881, 0.863641252539904754298979315776, 2.43226033432249507538981818591, 3.80124891073354771403603784450, 4.72125727212185564380413157815, 5.36592252478857706183298862512, 6.46217456101406926734272733058, 7.71749012162193003265448718997, 8.649182016249339130899834514543, 10.08268858497834225041654829348, 10.40838529098686602226147590593

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