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Properties

Label 2-624-208.187-c1-0-30
Degree $2$
Conductor $624$
Sign $0.326 - 0.945i$
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  + (−1.15 + 0.815i)2-s + (0.707 + 0.707i)3-s + (0.670 − 1.88i)4-s + 2.34·5-s + (−1.39 − 0.240i)6-s + (3.19 + 3.19i)7-s + (0.762 + 2.72i)8-s + 1.00i·9-s + (−2.71 + 1.91i)10-s + 5.07·11-s + (1.80 − 0.858i)12-s + (−3.40 − 1.19i)13-s + (−6.29 − 1.08i)14-s + (1.66 + 1.66i)15-s + (−3.10 − 2.52i)16-s − 3.26i·17-s + ⋯
L(s)  = 1  + (−0.817 + 0.576i)2-s + (0.408 + 0.408i)3-s + (0.335 − 0.942i)4-s + 1.05·5-s + (−0.568 − 0.0981i)6-s + (1.20 + 1.20i)7-s + (0.269 + 0.962i)8-s + 0.333i·9-s + (−0.858 + 0.605i)10-s + 1.52·11-s + (0.521 − 0.247i)12-s + (−0.943 − 0.330i)13-s + (−1.68 − 0.289i)14-s + (0.428 + 0.428i)15-s + (−0.775 − 0.631i)16-s − 0.792i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.326 - 0.945i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.326 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $0.326 - 0.945i$
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: \((2,\ 624,\ (\ :1/2),\ 0.326 - 0.945i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.30954 + 0.932820i\)
\(L(\frac12)\) \(\approx\) \(1.30954 + 0.932820i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.15 - 0.815i)T \)
3 \( 1 + (-0.707 - 0.707i)T \)
13 \( 1 + (3.40 + 1.19i)T \)
good5 \( 1 - 2.34T + 5T^{2} \)
7 \( 1 + (-3.19 - 3.19i)T + 7iT^{2} \)
11 \( 1 - 5.07T + 11T^{2} \)
17 \( 1 + 3.26iT - 17T^{2} \)
19 \( 1 + 0.605T + 19T^{2} \)
23 \( 1 + 2.94iT - 23T^{2} \)
29 \( 1 + (-0.887 + 0.887i)T - 29iT^{2} \)
31 \( 1 + (5.17 + 5.17i)T + 31iT^{2} \)
37 \( 1 - 7.71iT - 37T^{2} \)
41 \( 1 + (-1.94 - 1.94i)T + 41iT^{2} \)
43 \( 1 + (6.13 + 6.13i)T + 43iT^{2} \)
47 \( 1 + (-5.45 + 5.45i)T - 47iT^{2} \)
53 \( 1 + (-0.417 - 0.417i)T + 53iT^{2} \)
59 \( 1 + 12.6T + 59T^{2} \)
61 \( 1 + (5.83 - 5.83i)T - 61iT^{2} \)
67 \( 1 + 1.52T + 67T^{2} \)
71 \( 1 + (8.15 - 8.15i)T - 71iT^{2} \)
73 \( 1 + (-1.51 + 1.51i)T - 73iT^{2} \)
79 \( 1 - 1.03iT - 79T^{2} \)
83 \( 1 - 13.5T + 83T^{2} \)
89 \( 1 + (0.249 - 0.249i)T - 89iT^{2} \)
97 \( 1 + (-0.664 + 0.664i)T - 97iT^{2} \)
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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

−10.46053480193357409832263545394, −9.560488423798138097221060421950, −9.116684470796272227282651178143, −8.403205443734611668517169682307, −7.39012423081954484314707132357, −6.26079921499362102804393896153, −5.43693884709979009894398615969, −4.62744777908492202844614527525, −2.52273444046270133053430760972, −1.67535167236397369000380110159, 1.38709792954184927024868538534, 1.91510819959706694044633589114, 3.59078751123497575029849705704, 4.58402728271514227115810682100, 6.23901468455223126692212844352, 7.19405770548799816364478243665, 7.79578717689459251479455538035, 8.946660228939466511940057796152, 9.458467883740332476853164731014, 10.42224767411383220135865769243

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