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Properties

Label 2-624-208.181-c1-0-10
Degree $2$
Conductor $624$
Sign $0.926 - 0.377i$
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  + (0.144 − 1.40i)2-s + (−0.707 + 0.707i)3-s + (−1.95 − 0.405i)4-s + (0.341 − 0.341i)5-s + (0.892 + 1.09i)6-s − 3.12·7-s + (−0.852 + 2.69i)8-s − 1.00i·9-s + (−0.431 − 0.529i)10-s + (−2.20 + 2.20i)11-s + (1.67 − 1.09i)12-s + (3.31 − 1.41i)13-s + (−0.449 + 4.39i)14-s + 0.483i·15-s + (3.67 + 1.58i)16-s + 5.69·17-s + ⋯
L(s)  = 1  + (0.101 − 0.994i)2-s + (−0.408 + 0.408i)3-s + (−0.979 − 0.202i)4-s + (0.152 − 0.152i)5-s + (0.364 + 0.447i)6-s − 1.18·7-s + (−0.301 + 0.953i)8-s − 0.333i·9-s + (−0.136 − 0.167i)10-s + (−0.663 + 0.663i)11-s + (0.482 − 0.317i)12-s + (0.920 − 0.391i)13-s + (−0.120 + 1.17i)14-s + 0.124i·15-s + (0.917 + 0.396i)16-s + 1.38·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.926 - 0.377i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.926 - 0.377i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $0.926 - 0.377i$
Analytic conductor: \(4.98266\)
Root analytic conductor: \(2.23218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{624} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 624,\ (\ :1/2),\ 0.926 - 0.377i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.860317 + 0.168540i\)
\(L(\frac12)\) \(\approx\) \(0.860317 + 0.168540i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.144 + 1.40i)T \)
3 \( 1 + (0.707 - 0.707i)T \)
13 \( 1 + (-3.31 + 1.41i)T \)
good5 \( 1 + (-0.341 + 0.341i)T - 5iT^{2} \)
7 \( 1 + 3.12T + 7T^{2} \)
11 \( 1 + (2.20 - 2.20i)T - 11iT^{2} \)
17 \( 1 - 5.69T + 17T^{2} \)
19 \( 1 + (-3.24 - 3.24i)T + 19iT^{2} \)
23 \( 1 - 2.68iT - 23T^{2} \)
29 \( 1 + (3.40 - 3.40i)T - 29iT^{2} \)
31 \( 1 - 6.21iT - 31T^{2} \)
37 \( 1 + (6.35 - 6.35i)T - 37iT^{2} \)
41 \( 1 - 0.0996T + 41T^{2} \)
43 \( 1 + (-2.74 - 2.74i)T + 43iT^{2} \)
47 \( 1 + 12.8iT - 47T^{2} \)
53 \( 1 + (-5.81 - 5.81i)T + 53iT^{2} \)
59 \( 1 + (-1.03 + 1.03i)T - 59iT^{2} \)
61 \( 1 + (-8.75 + 8.75i)T - 61iT^{2} \)
67 \( 1 + (-6.24 - 6.24i)T + 67iT^{2} \)
71 \( 1 + 0.727T + 71T^{2} \)
73 \( 1 + 8.25T + 73T^{2} \)
79 \( 1 + 15.9T + 79T^{2} \)
83 \( 1 + (-3.36 - 3.36i)T + 83iT^{2} \)
89 \( 1 + 6.60T + 89T^{2} \)
97 \( 1 - 8.43iT - 97T^{2} \)
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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

−10.41887815193523241378512207521, −10.04657089179099152480603229001, −9.339698004111465315113987295467, −8.307424971236235637142530326885, −7.07934836085899133155963075263, −5.63810081232778065396032802027, −5.26952939019994086154908823474, −3.67034910986161357297505460284, −3.18620818988015232686382740501, −1.34512529869508071617546021757, 0.55164284161319892420947942969, 2.95014065038716802124126568307, 4.05272482509529411670919034676, 5.56701387147535963776517354524, 5.97779252842585640643813611500, 6.89302273602867395881305437004, 7.71529196897253902983470528495, 8.659457326378542643655583749433, 9.601921786184057559498734735067, 10.35732261765979986710192408912

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