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Properties

Label 2-624-39.32-c1-0-9
Degree $2$
Conductor $624$
Sign $-0.162 - 0.986i$
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  + (1.19 + 1.24i)3-s + (1.44 + 1.44i)5-s + (−0.918 + 3.42i)7-s + (−0.122 + 2.99i)9-s + (−0.0392 − 0.146i)11-s + (1.92 − 3.04i)13-s + (−0.0722 + 3.52i)15-s + (−1.60 − 2.78i)17-s + (−1.90 − 0.510i)19-s + (−5.38 + 2.96i)21-s + (−4.19 + 7.26i)23-s − 0.848i·25-s + (−3.89 + 3.44i)27-s + (−0.0238 − 0.0137i)29-s + (5.54 − 5.54i)31-s + ⋯
L(s)  = 1  + (0.692 + 0.721i)3-s + (0.644 + 0.644i)5-s + (−0.347 + 1.29i)7-s + (−0.0409 + 0.999i)9-s + (−0.0118 − 0.0441i)11-s + (0.533 − 0.845i)13-s + (−0.0186 + 0.911i)15-s + (−0.389 − 0.675i)17-s + (−0.437 − 0.117i)19-s + (−1.17 + 0.646i)21-s + (−0.875 + 1.51i)23-s − 0.169i·25-s + (−0.749 + 0.662i)27-s + (−0.00442 − 0.00255i)29-s + (0.996 − 0.996i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.162 - 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.162 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $-0.162 - 0.986i$
Analytic conductor: \(4.98266\)
Root analytic conductor: \(2.23218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{624} (305, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 624,\ (\ :1/2),\ -0.162 - 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.24411 + 1.46623i\)
\(L(\frac12)\) \(\approx\) \(1.24411 + 1.46623i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.19 - 1.24i)T \)
13 \( 1 + (-1.92 + 3.04i)T \)
good5 \( 1 + (-1.44 - 1.44i)T + 5iT^{2} \)
7 \( 1 + (0.918 - 3.42i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (0.0392 + 0.146i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (1.60 + 2.78i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.90 + 0.510i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (4.19 - 7.26i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.0238 + 0.0137i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-5.54 + 5.54i)T - 31iT^{2} \)
37 \( 1 + (-3.84 + 1.03i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-2.53 + 0.678i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-5.90 + 3.40i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.77 + 4.77i)T - 47iT^{2} \)
53 \( 1 - 13.3iT - 53T^{2} \)
59 \( 1 + (8.09 + 2.16i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-6.61 - 11.4i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.18 + 4.41i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (1.29 - 4.81i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (-6.22 - 6.22i)T + 73iT^{2} \)
79 \( 1 - 13.3T + 79T^{2} \)
83 \( 1 + (1.94 + 1.94i)T + 83iT^{2} \)
89 \( 1 + (2.87 + 10.7i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (17.7 + 4.75i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.66640144625958337251795092232, −9.837049394336435975693883643066, −9.245077630117206609400639940624, −8.424814085407584297504758879495, −7.46890158039594956044474003144, −6.03999662528368065703091126664, −5.58488678169407625975979558013, −4.16765394490812183089715036405, −2.90974536644603622971391054760, −2.30196337240086004388240561170, 1.02700036068765553660752008217, 2.19641477904689612243001825248, 3.72377320874589898996756030804, 4.55810697896127247213107206456, 6.27052170933290378445427373398, 6.63640804507723796136985146922, 7.83386426804437131886117619369, 8.585167085030796149580237267919, 9.404153169006577183610480966870, 10.24166017214088049512193748493

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