Properties

Label 2-504-63.25-c1-0-11
Degree $2$
Conductor $504$
Sign $0.0574 + 0.998i$
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  + (−1.69 − 0.332i)3-s + (−1.59 + 2.76i)5-s + (−1.66 − 2.05i)7-s + (2.77 + 1.12i)9-s + (1.14 + 1.97i)11-s + (−0.675 − 1.16i)13-s + (3.63 − 4.17i)15-s + (2.21 − 3.83i)17-s + (−3.69 − 6.39i)19-s + (2.15 + 4.04i)21-s + (3.23 − 5.60i)23-s + (−2.60 − 4.51i)25-s + (−4.34 − 2.84i)27-s + (−1.06 + 1.83i)29-s − 0.632·31-s + ⋯
L(s)  = 1  + (−0.981 − 0.191i)3-s + (−0.714 + 1.23i)5-s + (−0.629 − 0.776i)7-s + (0.926 + 0.376i)9-s + (0.344 + 0.596i)11-s + (−0.187 − 0.324i)13-s + (0.938 − 1.07i)15-s + (0.537 − 0.930i)17-s + (−0.847 − 1.46i)19-s + (0.469 + 0.883i)21-s + (0.674 − 1.16i)23-s + (−0.520 − 0.902i)25-s + (−0.837 − 0.547i)27-s + (−0.197 + 0.341i)29-s − 0.113·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0574 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0574 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.0574 + 0.998i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.0574 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.400515 - 0.378140i\)
\(L(\frac12)\) \(\approx\) \(0.400515 - 0.378140i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.69 + 0.332i)T \)
7 \( 1 + (1.66 + 2.05i)T \)
good5 \( 1 + (1.59 - 2.76i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.14 - 1.97i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.675 + 1.16i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.21 + 3.83i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.69 + 6.39i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.23 + 5.60i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.06 - 1.83i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 0.632T + 31T^{2} \)
37 \( 1 + (-1.92 - 3.34i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (5.05 + 8.74i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.24 + 7.35i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 6.53T + 47T^{2} \)
53 \( 1 + (-2.39 + 4.15i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 6.20T + 59T^{2} \)
61 \( 1 + 8.91T + 61T^{2} \)
67 \( 1 + 3.01T + 67T^{2} \)
71 \( 1 + 15.3T + 71T^{2} \)
73 \( 1 + (-4.36 + 7.56i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 1.87T + 79T^{2} \)
83 \( 1 + (3.00 - 5.19i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-2.65 - 4.59i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-7.44 + 12.8i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.60734614187305001200910987486, −10.32710496319565119238099272986, −9.052623286633932379181224745948, −7.38542743246415615185616692001, −7.10135943968657212175815674522, −6.40039193388994774690357033400, −4.95701913698084638314070314718, −3.94878016231332267441736921256, −2.68738431903723794298502735368, −0.40803887020019936189085156087, 1.34857900236177476670164936704, 3.60432747781395724358817002909, 4.47930097780997414079952118818, 5.67235909238679396863273169059, 6.14814662158630921627409919188, 7.58221428783350969967007775894, 8.559627171325666362293961067172, 9.339890238897469901674111485211, 10.24450354304030872527781526391, 11.38580237816844031948356881694

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