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Properties

Label 2-504-63.41-c1-0-7
Degree $2$
Conductor $504$
Sign $0.951 - 0.307i$
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  + (−1.50 − 0.855i)3-s + (0.422 − 0.731i)5-s + (0.327 + 2.62i)7-s + (1.53 + 2.57i)9-s + (−0.791 + 0.456i)11-s + (−0.472 − 0.272i)13-s + (−1.26 + 0.740i)15-s + 4.77·17-s + 3.15i·19-s + (1.75 − 4.23i)21-s + (1.39 + 0.804i)23-s + (2.14 + 3.71i)25-s + (−0.108 − 5.19i)27-s + (5.56 − 3.21i)29-s + (1.57 + 0.908i)31-s + ⋯
L(s)  = 1  + (−0.869 − 0.493i)3-s + (0.188 − 0.327i)5-s + (0.123 + 0.992i)7-s + (0.512 + 0.858i)9-s + (−0.238 + 0.137i)11-s + (−0.131 − 0.0756i)13-s + (−0.325 + 0.191i)15-s + 1.15·17-s + 0.723i·19-s + (0.382 − 0.923i)21-s + (0.290 + 0.167i)23-s + (0.428 + 0.742i)25-s + (−0.0209 − 0.999i)27-s + (1.03 − 0.596i)29-s + (0.282 + 0.163i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.951 - 0.307i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.951 - 0.307i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.951 - 0.307i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.951 - 0.307i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.10238 + 0.173759i\)
\(L(\frac12)\) \(\approx\) \(1.10238 + 0.173759i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.50 + 0.855i)T \)
7 \( 1 + (-0.327 - 2.62i)T \)
good5 \( 1 + (-0.422 + 0.731i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.791 - 0.456i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.472 + 0.272i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 4.77T + 17T^{2} \)
19 \( 1 - 3.15iT - 19T^{2} \)
23 \( 1 + (-1.39 - 0.804i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-5.56 + 3.21i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.57 - 0.908i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 7.14T + 37T^{2} \)
41 \( 1 + (-2.82 + 4.88i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.84 + 3.19i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-6.75 - 11.6i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 2.49iT - 53T^{2} \)
59 \( 1 + (-0.279 + 0.483i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (10.9 - 6.34i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.06 - 5.30i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 9.24iT - 71T^{2} \)
73 \( 1 - 8.87iT - 73T^{2} \)
79 \( 1 + (5.58 + 9.67i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.122 - 0.211i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 6.19T + 89T^{2} \)
97 \( 1 + (12.2 - 7.06i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.05355570682181013635951802320, −10.16223665542641637542684549632, −9.244653301468885830477464050539, −8.131725191042508466107835215672, −7.36647709664157895794268904622, −6.05752902696286090130591356758, −5.55326169898209785657251458140, −4.55092755817128602344826568279, −2.75384995387511535822843082301, −1.31217942630439241055229797243, 0.895693045943934778568224129062, 3.03702566525143161847643641597, 4.29038031614235924867761952729, 5.12852278637168364776035702558, 6.29056104194951736729473889351, 7.03871004048675293882155732495, 8.084714765373955314159426706139, 9.399933334165890623917417094760, 10.24157246278253408273521277014, 10.71391585881554903452298616532

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