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Properties

Label 2-624-13.10-c1-0-8
Degree $2$
Conductor $624$
Sign $0.985 - 0.171i$
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 − 1.55i·5-s + (2.56 + 1.48i)7-s + (−0.499 + 0.866i)9-s + (1.94 − 1.12i)11-s + (−0.663 − 3.54i)13-s + (1.34 − 0.777i)15-s + (−0.509 + 0.882i)17-s + (4.63 + 2.67i)19-s + 2.96i·21-s + (−0.391 − 0.678i)23-s + 2.58·25-s − 0.999·27-s + (−1.34 − 2.33i)29-s + 7.28i·31-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s − 0.695i·5-s + (0.969 + 0.559i)7-s + (−0.166 + 0.288i)9-s + (0.586 − 0.338i)11-s + (−0.183 − 0.982i)13-s + (0.347 − 0.200i)15-s + (−0.123 + 0.213i)17-s + (1.06 + 0.614i)19-s + 0.646i·21-s + (−0.0816 − 0.141i)23-s + 0.516·25-s − 0.192·27-s + (−0.249 − 0.433i)29-s + 1.30i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.171i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 - 0.171i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $0.985 - 0.171i$
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: \((2,\ 624,\ (\ :1/2),\ 0.985 - 0.171i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.85204 + 0.159788i\)
\(L(\frac12)\) \(\approx\) \(1.85204 + 0.159788i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.663 + 3.54i)T \)
good5 \( 1 + 1.55iT - 5T^{2} \)
7 \( 1 + (-2.56 - 1.48i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.94 + 1.12i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (0.509 - 0.882i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.63 - 2.67i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.391 + 0.678i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.34 + 2.33i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 7.28iT - 31T^{2} \)
37 \( 1 + (-6.75 + 3.90i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (4.62 - 2.67i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.56 - 2.70i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 7.13iT - 47T^{2} \)
53 \( 1 - 13.8T + 53T^{2} \)
59 \( 1 + (-3.77 - 2.17i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.57 - 4.45i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (11.2 - 6.49i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (11.9 + 6.92i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 8.30iT - 73T^{2} \)
79 \( 1 + 1.23T + 79T^{2} \)
83 \( 1 - 7.67iT - 83T^{2} \)
89 \( 1 + (0.427 - 0.247i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (9.61 + 5.55i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.55493223232892871325094540356, −9.728771894951254265468383125959, −8.679856512151505790988171802357, −8.359140773148429384444146693060, −7.28117761286395916358247696038, −5.74718955642886847781571195840, −5.15848020657420521923039672744, −4.12364421207989336134466357988, −2.88582282231419311396904251133, −1.33897450778188244624239547141, 1.37376340165570619018394516555, 2.62187125000929702162564890936, 3.95990501570497170854969339426, 4.94598536138867554427956007346, 6.34254315969424108852252262797, 7.18748793615816733659502945382, 7.67463147602970347422958479287, 8.876280231311572646130325809119, 9.628125102768826248869999572001, 10.71780402785583890063450015504

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