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Properties

Label 2-624-13.4-c1-0-1
Degree $2$
Conductor $624$
Sign $-0.702 - 0.711i$
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 + 3.73i·5-s + (−2.36 + 1.36i)7-s + (−0.499 − 0.866i)9-s + (−1.09 − 0.633i)11-s + (−2.59 + 2.5i)13-s + (3.23 + 1.86i)15-s + (−2.86 − 4.96i)17-s + (−4.09 + 2.36i)19-s + 2.73i·21-s + (−2.09 + 3.63i)23-s − 8.92·25-s − 0.999·27-s + (2.23 − 3.86i)29-s − 1.46i·31-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s + 1.66i·5-s + (−0.894 + 0.516i)7-s + (−0.166 − 0.288i)9-s + (−0.331 − 0.191i)11-s + (−0.720 + 0.693i)13-s + (0.834 + 0.481i)15-s + (−0.695 − 1.20i)17-s + (−0.940 + 0.542i)19-s + 0.596i·21-s + (−0.437 + 0.757i)23-s − 1.78·25-s − 0.192·27-s + (0.414 − 0.717i)29-s − 0.262i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.702 - 0.711i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.702 - 0.711i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $-0.702 - 0.711i$
Analytic conductor: \(4.98266\)
Root analytic conductor: \(2.23218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{624} (433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 624,\ (\ :1/2),\ -0.702 - 0.711i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.305857 + 0.731761i\)
\(L(\frac12)\) \(\approx\) \(0.305857 + 0.731761i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.59 - 2.5i)T \)
good5 \( 1 - 3.73iT - 5T^{2} \)
7 \( 1 + (2.36 - 1.36i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.09 + 0.633i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (2.86 + 4.96i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.09 - 2.36i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.09 - 3.63i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.23 + 3.86i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 1.46iT - 31T^{2} \)
37 \( 1 + (-3.06 - 1.76i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-8.13 - 4.69i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.83 - 8.36i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 2.19iT - 47T^{2} \)
53 \( 1 + 6.46T + 53T^{2} \)
59 \( 1 + (-6.92 + 4i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.59 - 7.96i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-11.3 - 6.56i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (4.09 - 2.36i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 6.26iT - 73T^{2} \)
79 \( 1 - 2.53T + 79T^{2} \)
83 \( 1 - 0.196iT - 83T^{2} \)
89 \( 1 + (8.19 + 4.73i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.19 - 3i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.09183410343291506671361223744, −9.846492301654496377446547903521, −9.493213099772387690084741637890, −8.126006971168066879377349107495, −7.24021351661811350938165437971, −6.55623282563754418818297559008, −5.89686578922987762295788452171, −4.18557868460119040648143612469, −2.83950675583261077305839377604, −2.42272070900875439720971492115, 0.38706085659547321548333630484, 2.27895900749039203450240028767, 3.84973686827284341600959328169, 4.58437145275274747665279125953, 5.50608328111035772300690237177, 6.65817814973386917527968695676, 7.926404970310890597295311601401, 8.653634188402530987474949157961, 9.329390106688636734482775811878, 10.22500011470123770682811914148

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