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Properties

Label 2-624-39.2-c1-0-20
Degree $2$
Conductor $624$
Sign $-0.595 + 0.803i$
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  + (−0.852 + 1.50i)3-s + (0.477 + 0.477i)5-s + (−0.988 + 0.264i)7-s + (−1.54 − 2.57i)9-s + (−5.10 − 1.36i)11-s + (−1.49 − 3.28i)13-s + (−1.12 + 0.312i)15-s + (−1.22 + 2.11i)17-s + (1.14 + 4.25i)19-s + (0.443 − 1.71i)21-s + (−2.40 − 4.17i)23-s − 4.54i·25-s + (5.19 − 0.137i)27-s + (−3.23 + 1.86i)29-s + (2.91 − 2.91i)31-s + ⋯
L(s)  = 1  + (−0.492 + 0.870i)3-s + (0.213 + 0.213i)5-s + (−0.373 + 0.100i)7-s + (−0.515 − 0.857i)9-s + (−1.53 − 0.412i)11-s + (−0.413 − 0.910i)13-s + (−0.291 + 0.0807i)15-s + (−0.296 + 0.513i)17-s + (0.261 + 0.976i)19-s + (0.0968 − 0.374i)21-s + (−0.502 − 0.870i)23-s − 0.908i·25-s + (0.999 − 0.0263i)27-s + (−0.600 + 0.346i)29-s + (0.523 − 0.523i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.595 + 0.803i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.595 + 0.803i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $-0.595 + 0.803i$
Analytic conductor: \(4.98266\)
Root analytic conductor: \(2.23218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{624} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 624,\ (\ :1/2),\ -0.595 + 0.803i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0587158 - 0.116627i\)
\(L(\frac12)\) \(\approx\) \(0.0587158 - 0.116627i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.852 - 1.50i)T \)
13 \( 1 + (1.49 + 3.28i)T \)
good5 \( 1 + (-0.477 - 0.477i)T + 5iT^{2} \)
7 \( 1 + (0.988 - 0.264i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (5.10 + 1.36i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (1.22 - 2.11i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.14 - 4.25i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (2.40 + 4.17i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.23 - 1.86i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.91 + 2.91i)T - 31iT^{2} \)
37 \( 1 + (0.898 - 3.35i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-1.25 + 4.66i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (10.9 + 6.33i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.646 + 0.646i)T - 47iT^{2} \)
53 \( 1 - 6.56iT - 53T^{2} \)
59 \( 1 + (2.21 + 8.28i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (3.99 - 6.92i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.54 + 0.949i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-5.34 + 1.43i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (0.225 + 0.225i)T + 73iT^{2} \)
79 \( 1 - 2.70T + 79T^{2} \)
83 \( 1 + (-7.41 - 7.41i)T + 83iT^{2} \)
89 \( 1 + (10.0 + 2.68i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (2.57 + 9.59i)T + (-84.0 + 48.5i)T^{2} \)
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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

−10.37137074815377325492914325329, −9.790646198101288826184802034334, −8.541348404125912502005002871030, −7.83160070746855782262056787434, −6.40793742277014971128010634663, −5.68155524061845780700821655949, −4.86479641431027595178825020133, −3.58781312737727576178856563413, −2.57521672656063232050431658331, −0.07030744026698571678216774995, 1.81486383204990047429309684003, 2.93733045289624725518952504642, 4.76371872419774214848649786224, 5.40001629759008987528754232220, 6.57801954219467314394923651540, 7.30687517791784631087693281564, 8.052468772528976345606132431346, 9.273274753266521226336731922679, 10.01636654392056986206252100495, 11.12649393643096590215982362344

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