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Properties

Label 2-504-63.59-c1-0-16
Degree $2$
Conductor $504$
Sign $0.0332 + 0.999i$
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  + (−0.210 + 1.71i)3-s − 1.07·5-s + (−1.26 − 2.32i)7-s + (−2.91 − 0.724i)9-s − 4.09i·11-s + (−3.69 − 2.13i)13-s + (0.226 − 1.84i)15-s + (0.717 − 1.24i)17-s + (6.41 − 3.70i)19-s + (4.25 − 1.68i)21-s + 6.27i·23-s − 3.84·25-s + (1.85 − 4.85i)27-s + (−8.09 + 4.67i)29-s + (5.96 − 3.44i)31-s + ⋯
L(s)  = 1  + (−0.121 + 0.992i)3-s − 0.480·5-s + (−0.478 − 0.877i)7-s + (−0.970 − 0.241i)9-s − 1.23i·11-s + (−1.02 − 0.592i)13-s + (0.0584 − 0.477i)15-s + (0.174 − 0.301i)17-s + (1.47 − 0.850i)19-s + (0.929 − 0.368i)21-s + 1.30i·23-s − 0.768·25-s + (0.357 − 0.933i)27-s + (−1.50 + 0.868i)29-s + (1.07 − 0.618i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0332 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0332 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.0332 + 0.999i$
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: \((2,\ 504,\ (\ :1/2),\ 0.0332 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.467241 - 0.451961i\)
\(L(\frac12)\) \(\approx\) \(0.467241 - 0.451961i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.210 - 1.71i)T \)
7 \( 1 + (1.26 + 2.32i)T \)
good5 \( 1 + 1.07T + 5T^{2} \)
11 \( 1 + 4.09iT - 11T^{2} \)
13 \( 1 + (3.69 + 2.13i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.717 + 1.24i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-6.41 + 3.70i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 6.27iT - 23T^{2} \)
29 \( 1 + (8.09 - 4.67i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-5.96 + 3.44i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.453 + 0.785i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.88 + 6.73i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (6.32 + 10.9i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.21 - 7.30i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.50 - 0.869i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.05 + 5.28i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.36 + 1.36i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.01 + 10.4i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 0.783iT - 71T^{2} \)
73 \( 1 + (1.95 + 1.13i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.817 - 1.41i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.48 - 7.77i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.71 - 2.97i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.05 + 2.91i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.75665016123042933976635831654, −9.729774627742352103999192410798, −9.257760692468942906399305594860, −7.914658039892192568425453464811, −7.22995753049767543430882336784, −5.78574457842729362664244041942, −5.01529719532440387993219962772, −3.69661844229306612497113834892, −3.12851497245345628816601360312, −0.38121208046182738174114472273, 1.84024896978949631251806987528, 2.95449255081931658887558052335, 4.54981977594852128300855922456, 5.69081256675846195927687391558, 6.65342640406771958521281812674, 7.51958273712984234537908696050, 8.205625701057835436074314577300, 9.437891870402434272948858266518, 10.05778595934909647383405986768, 11.75876381781896867035399036759

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