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Properties

Label 2-624-208.19-c1-0-28
Degree $2$
Conductor $624$
Sign $0.729 - 0.684i$
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  + (1.37 + 0.349i)2-s + (−0.258 − 0.965i)3-s + (1.75 + 0.958i)4-s + 0.533i·5-s + (−0.0168 − 1.41i)6-s + (−0.607 + 2.26i)7-s + (2.07 + 1.92i)8-s + (−0.866 + 0.499i)9-s + (−0.186 + 0.731i)10-s + (3.37 + 1.94i)11-s + (0.471 − 1.94i)12-s + (−2.94 + 2.07i)13-s + (−1.62 + 2.89i)14-s + (0.515 − 0.138i)15-s + (2.16 + 3.36i)16-s + (3.15 − 1.81i)17-s + ⋯
L(s)  = 1  + (0.968 + 0.247i)2-s + (−0.149 − 0.557i)3-s + (0.877 + 0.479i)4-s + 0.238i·5-s + (−0.00689 − 0.577i)6-s + (−0.229 + 0.857i)7-s + (0.731 + 0.681i)8-s + (−0.288 + 0.166i)9-s + (−0.0590 + 0.231i)10-s + (1.01 + 0.587i)11-s + (0.136 − 0.561i)12-s + (−0.817 + 0.575i)13-s + (−0.434 + 0.773i)14-s + (0.133 − 0.0356i)15-s + (0.540 + 0.841i)16-s + (0.764 − 0.441i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.729 - 0.684i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.729 - 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $0.729 - 0.684i$
Analytic conductor: \(4.98266\)
Root analytic conductor: \(2.23218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{624} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 624,\ (\ :1/2),\ 0.729 - 0.684i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.40328 + 0.950955i\)
\(L(\frac12)\) \(\approx\) \(2.40328 + 0.950955i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.37 - 0.349i)T \)
3 \( 1 + (0.258 + 0.965i)T \)
13 \( 1 + (2.94 - 2.07i)T \)
good5 \( 1 - 0.533iT - 5T^{2} \)
7 \( 1 + (0.607 - 2.26i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (-3.37 - 1.94i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-3.15 + 1.81i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.70 + 0.986i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.31 + 0.757i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.79 - 0.481i)T + (25.1 - 14.5i)T^{2} \)
31 \( 1 + (6.22 + 6.22i)T + 31iT^{2} \)
37 \( 1 + (-5.28 + 9.15i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.24 - 0.600i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-0.913 - 0.244i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (8.16 - 8.16i)T - 47iT^{2} \)
53 \( 1 + (-6.50 + 6.50i)T - 53iT^{2} \)
59 \( 1 + (5.15 - 2.97i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.62 + 9.80i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (-2.24 - 1.29i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (15.2 + 4.09i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (3.70 - 3.70i)T - 73iT^{2} \)
79 \( 1 - 0.388iT - 79T^{2} \)
83 \( 1 - 12.6iT - 83T^{2} \)
89 \( 1 + (3.89 + 14.5i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-2.20 - 0.590i)T + (84.0 + 48.5i)T^{2} \)
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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

−11.19913870857477371681336106873, −9.778914015966815655974397825358, −8.973974931780673250637102043050, −7.62843550762579597262629401190, −7.05619361510701474134648625101, −6.16214008056939776662713154204, −5.33535999885445578590436986506, −4.24110845642839136352066540577, −2.94227036262201175749879059429, −1.91179445685313259083603686517, 1.20862548005851772024922768184, 3.13061586659043302896046807676, 3.83790847945946440400143198226, 4.86961586473099061374489435544, 5.73205655642289168978594834612, 6.73383768079863037306043275666, 7.63632578390987102520784814460, 8.925560263084409528287398002732, 10.07926951143692833003209356057, 10.42029938493106382199509775020

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