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Properties

Label 2-504-63.59-c1-0-18
Degree $2$
Conductor $504$
Sign $-0.120 + 0.992i$
Analytic cond. $4.02446$
Root an. cond. $2.00610$
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  + (0.549 − 1.64i)3-s − 1.05·5-s + (1.79 + 1.94i)7-s + (−2.39 − 1.80i)9-s − 6.24i·11-s + (−0.872 − 0.503i)13-s + (−0.580 + 1.73i)15-s + (3.26 − 5.66i)17-s + (1.73 − 1.00i)19-s + (4.17 − 1.88i)21-s + 4.40i·23-s − 3.88·25-s + (−4.28 + 2.94i)27-s + (6.12 − 3.53i)29-s + (−2.07 + 1.19i)31-s + ⋯
L(s)  = 1  + (0.317 − 0.948i)3-s − 0.472·5-s + (0.679 + 0.733i)7-s + (−0.798 − 0.601i)9-s − 1.88i·11-s + (−0.241 − 0.139i)13-s + (−0.149 + 0.447i)15-s + (0.792 − 1.37i)17-s + (0.397 − 0.229i)19-s + (0.911 − 0.412i)21-s + 0.917i·23-s − 0.777·25-s + (−0.823 + 0.566i)27-s + (1.13 − 0.657i)29-s + (−0.372 + 0.214i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.120 + 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.120 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.120 + 0.992i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (185, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ -0.120 + 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.942123 - 1.06304i\)
\(L(\frac12)\) \(\approx\) \(0.942123 - 1.06304i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.549 + 1.64i)T \)
7 \( 1 + (-1.79 - 1.94i)T \)
good5 \( 1 + 1.05T + 5T^{2} \)
11 \( 1 + 6.24iT - 11T^{2} \)
13 \( 1 + (0.872 + 0.503i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.26 + 5.66i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.73 + 1.00i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 4.40iT - 23T^{2} \)
29 \( 1 + (-6.12 + 3.53i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.07 - 1.19i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.64 + 6.30i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.80 + 3.11i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.60 - 2.78i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.87 - 3.23i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.02 + 3.47i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6.67 - 11.5i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7.10 - 4.10i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.0613 + 0.106i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 5.37iT - 71T^{2} \)
73 \( 1 + (-14.4 - 8.33i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.43 + 7.67i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.07 + 1.86i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (2.23 + 3.86i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.960 + 0.554i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.08873575593972763033088439934, −9.538127743550106296585230684159, −8.667928014116660337242390999110, −7.982809907097639153695172115591, −7.27116224209940219474890449002, −5.94389003911521549119713218388, −5.29605164717438154804979033177, −3.51401328337232741082419300677, −2.57008985019081592971799667174, −0.862574353254291662659081379393, 1.91932828543638352957154136255, 3.58894934277232035241749016693, 4.42667761365119319521431493578, 5.12677092603993932644837097738, 6.70998595882647498740051361041, 7.80566074871363726797465153286, 8.312492142523554436721316190183, 9.675306301092230775082415478746, 10.17641261039920825538114630143, 10.93056223157900974764428339831

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