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Properties

Label 2-624-208.171-c1-0-20
Degree $2$
Conductor $624$
Sign $-0.175 + 0.984i$
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  + (−0.557 − 1.29i)2-s + (−0.258 + 0.965i)3-s + (−1.37 + 1.44i)4-s − 3.18·5-s + (1.39 − 0.202i)6-s + (−1.75 + 0.470i)7-s + (2.65 + 0.983i)8-s + (−0.866 − 0.499i)9-s + (1.77 + 4.14i)10-s + (2.49 + 4.31i)11-s + (−1.04 − 1.70i)12-s + (0.250 − 3.59i)13-s + (1.59 + 2.01i)14-s + (0.825 − 3.07i)15-s + (−0.199 − 3.99i)16-s + (−0.0307 − 0.0177i)17-s + ⋯
L(s)  = 1  + (−0.394 − 0.919i)2-s + (−0.149 + 0.557i)3-s + (−0.689 + 0.724i)4-s − 1.42·5-s + (0.571 − 0.0824i)6-s + (−0.663 + 0.177i)7-s + (0.937 + 0.347i)8-s + (−0.288 − 0.166i)9-s + (0.561 + 1.31i)10-s + (0.751 + 1.30i)11-s + (−0.301 − 0.492i)12-s + (0.0695 − 0.997i)13-s + (0.425 + 0.539i)14-s + (0.213 − 0.795i)15-s + (−0.0498 − 0.998i)16-s + (−0.00746 − 0.00430i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.175 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.175 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $-0.175 + 0.984i$
Analytic conductor: \(4.98266\)
Root analytic conductor: \(2.23218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{624} (379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 624,\ (\ :1/2),\ -0.175 + 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.339260 - 0.405085i\)
\(L(\frac12)\) \(\approx\) \(0.339260 - 0.405085i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.557 + 1.29i)T \)
3 \( 1 + (0.258 - 0.965i)T \)
13 \( 1 + (-0.250 + 3.59i)T \)
good5 \( 1 + 3.18T + 5T^{2} \)
7 \( 1 + (1.75 - 0.470i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (-2.49 - 4.31i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (0.0307 + 0.0177i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.25 + 5.64i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.42 - 1.40i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (5.57 + 1.49i)T + (25.1 + 14.5i)T^{2} \)
31 \( 1 + (3.84 + 3.84i)T + 31iT^{2} \)
37 \( 1 + (-7.15 + 4.13i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.83 + 10.5i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-12.0 + 3.22i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (-4.35 + 4.35i)T - 47iT^{2} \)
53 \( 1 + (-6.42 - 6.42i)T + 53iT^{2} \)
59 \( 1 + (-0.00934 + 0.0161i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.759 - 2.83i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (4.23 + 7.33i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (0.0593 + 0.221i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (6.25 - 6.25i)T - 73iT^{2} \)
79 \( 1 + 3.65iT - 79T^{2} \)
83 \( 1 - 11.6T + 83T^{2} \)
89 \( 1 + (7.45 + 1.99i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-1.18 - 4.44i)T + (-84.0 + 48.5i)T^{2} \)
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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

−10.48315260982229703631912990649, −9.390600634447678980372960356676, −9.093130875828208365423694039248, −7.66710422717885685848953111852, −7.29566027783563544572092125712, −5.53780679981057013174682705627, −4.23207643162455613557955874644, −3.79937167934781843302745915903, −2.57987178331745656310399620498, −0.42955266922518680885119176194, 1.08398116550722466761362283445, 3.49557255287625179338622250860, 4.27315355003060456608063421080, 5.80933433073562982579097791811, 6.47078620457894141408442208674, 7.41157443871701528417876557793, 8.023102323323430389838903067965, 8.869377585705966375246137132674, 9.704111044990576123081460967586, 11.01121126118761198637014662096

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