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Properties

Label 2-624-208.173-c1-0-14
Degree $2$
Conductor $624$
Sign $0.783 + 0.620i$
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  + (−1.28 − 0.581i)2-s + (−0.965 + 0.258i)3-s + (1.32 + 1.49i)4-s + (−0.295 − 0.295i)5-s + (1.39 + 0.228i)6-s + (−0.121 − 0.210i)7-s + (−0.833 − 2.70i)8-s + (0.866 − 0.499i)9-s + (0.208 + 0.552i)10-s + (2.98 − 0.799i)11-s + (−1.66 − 1.10i)12-s + (−2.29 + 2.78i)13-s + (0.0342 + 0.342i)14-s + (0.361 + 0.208i)15-s + (−0.497 + 3.96i)16-s + (−1.37 − 2.38i)17-s + ⋯
L(s)  = 1  + (−0.911 − 0.411i)2-s + (−0.557 + 0.149i)3-s + (0.661 + 0.749i)4-s + (−0.131 − 0.131i)5-s + (0.569 + 0.0931i)6-s + (−0.0460 − 0.0797i)7-s + (−0.294 − 0.955i)8-s + (0.288 − 0.166i)9-s + (0.0660 + 0.174i)10-s + (0.900 − 0.241i)11-s + (−0.481 − 0.319i)12-s + (−0.635 + 0.772i)13-s + (0.00916 + 0.0915i)14-s + (0.0933 + 0.0538i)15-s + (−0.124 + 0.992i)16-s + (−0.334 − 0.578i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.783 + 0.620i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.783 + 0.620i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $0.783 + 0.620i$
Analytic conductor: \(4.98266\)
Root analytic conductor: \(2.23218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{624} (589, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 624,\ (\ :1/2),\ 0.783 + 0.620i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.728528 - 0.253516i\)
\(L(\frac12)\) \(\approx\) \(0.728528 - 0.253516i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.28 + 0.581i)T \)
3 \( 1 + (0.965 - 0.258i)T \)
13 \( 1 + (2.29 - 2.78i)T \)
good5 \( 1 + (0.295 + 0.295i)T + 5iT^{2} \)
7 \( 1 + (0.121 + 0.210i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.98 + 0.799i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (1.37 + 2.38i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.59 - 0.426i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (2.23 + 1.29i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-9.23 + 2.47i)T + (25.1 - 14.5i)T^{2} \)
31 \( 1 - 11.0iT - 31T^{2} \)
37 \( 1 + (-7.08 + 1.89i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-2.47 + 4.28i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.20 - 0.859i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 - 4.60iT - 47T^{2} \)
53 \( 1 + (-6.92 + 6.92i)T - 53iT^{2} \)
59 \( 1 + (-1.94 + 7.27i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-2.52 + 9.42i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (-0.638 - 2.38i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (2.77 + 4.81i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 2.41T + 73T^{2} \)
79 \( 1 - 17.0T + 79T^{2} \)
83 \( 1 + (-3.13 + 3.13i)T - 83iT^{2} \)
89 \( 1 + (7.94 - 13.7i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.94 - 1.12i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.45326011536882657332050530591, −9.687805343697480131436742174369, −8.948838959819758272569262906825, −8.049837274049032893555082782100, −6.88389265592302423155796433238, −6.40118084081469335762992151567, −4.82055361867621711738197402489, −3.81643327201984632689766878292, −2.38230794635496290681065342964, −0.813348980596395669593674226569, 1.03251589737118955665655063829, 2.57243612696630902055529254060, 4.30684483728427321851220228329, 5.58332215380006713502951715189, 6.30181905512712790414564058728, 7.26073918634379501601372364216, 7.930882473030385933088929735432, 9.025913592255078610116238789278, 9.822363239547698077790848548442, 10.54510741243998495691678274767

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