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Properties

Label 2-504-7.2-c1-0-1
Degree $2$
Conductor $504$
Sign $-0.386 - 0.922i$
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  + (−2 + 3.46i)5-s + (2.5 − 0.866i)7-s − 3·13-s + (2 + 3.46i)17-s + (−3.5 + 6.06i)19-s + (−2 + 3.46i)23-s + (−5.49 − 9.52i)25-s − 8·29-s + (2.5 + 4.33i)31-s + (−2.00 + 10.3i)35-s + (−1.5 + 2.59i)37-s + 8·41-s + 11·43-s + (−2 + 3.46i)47-s + (5.5 − 4.33i)49-s + ⋯
L(s)  = 1  + (−0.894 + 1.54i)5-s + (0.944 − 0.327i)7-s − 0.832·13-s + (0.485 + 0.840i)17-s + (−0.802 + 1.39i)19-s + (−0.417 + 0.722i)23-s + (−1.09 − 1.90i)25-s − 1.48·29-s + (0.449 + 0.777i)31-s + (−0.338 + 1.75i)35-s + (−0.246 + 0.427i)37-s + 1.24·41-s + 1.67·43-s + (−0.291 + 0.505i)47-s + (0.785 − 0.618i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.386 - 0.922i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ -0.386 - 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.574728 + 0.864016i\)
\(L(\frac12)\) \(\approx\) \(0.574728 + 0.864016i\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 + (-2.5 + 0.866i)T \)
good5 \( 1 + (2 - 3.46i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 3T + 13T^{2} \)
17 \( 1 + (-2 - 3.46i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.5 - 6.06i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2 - 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 8T + 29T^{2} \)
31 \( 1 + (-2.5 - 4.33i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.5 - 2.59i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 8T + 41T^{2} \)
43 \( 1 - 11T + 43T^{2} \)
47 \( 1 + (2 - 3.46i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2 + 3.46i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (6 + 10.3i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1 + 1.73i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.5 - 2.59i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 + (0.5 + 0.866i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 12T + 83T^{2} \)
89 \( 1 + (4 - 6.92i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 2T + 97T^{2} \)
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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

−10.98924161734576861982342676356, −10.62010952116005021816092700862, −9.637049345552500459431017087019, −7.966977003140255282520592127949, −7.82275404533879712100488741639, −6.77641620994493579268133348599, −5.70776048176978650128831836495, −4.24230761939033025582636775265, −3.46278771705492276316665914800, −2.02300429895265815354099123939, 0.61964257847577541227096568790, 2.32856040431564032607720466055, 4.21128446977269884399013820835, 4.77501246160384940741830682600, 5.66617926374476197732742632562, 7.36748687500855343626696439633, 7.927506598256312011528044874377, 8.915407422879308146679300507267, 9.385722123568035517305389712031, 10.90159824313661073380379620450

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