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Properties

Label 2-504-168.107-c1-0-25
Degree $2$
Conductor $504$
Sign $-0.661 + 0.750i$
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  + (−1.11 − 0.874i)2-s + (0.471 + 1.94i)4-s + (1.51 − 2.62i)5-s + (−1.48 − 2.18i)7-s + (1.17 − 2.57i)8-s + (−3.97 + 1.59i)10-s + (4.18 − 2.41i)11-s + 1.60i·13-s + (−0.255 + 3.73i)14-s + (−3.55 + 1.83i)16-s + (−5.79 + 3.34i)17-s + (−0.663 + 1.14i)19-s + (5.81 + 1.70i)20-s + (−6.75 − 0.971i)22-s + (4.32 − 7.49i)23-s + ⋯
L(s)  = 1  + (−0.786 − 0.618i)2-s + (0.235 + 0.971i)4-s + (0.677 − 1.17i)5-s + (−0.563 − 0.826i)7-s + (0.415 − 0.909i)8-s + (−1.25 + 0.503i)10-s + (1.26 − 0.727i)11-s + 0.445i·13-s + (−0.0683 + 0.997i)14-s + (−0.888 + 0.458i)16-s + (−1.40 + 0.812i)17-s + (−0.152 + 0.263i)19-s + (1.30 + 0.381i)20-s + (−1.44 − 0.207i)22-s + (0.902 − 1.56i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.661 + 0.750i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.661 + 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.661 + 0.750i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ -0.661 + 0.750i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.394085 - 0.872510i\)
\(L(\frac12)\) \(\approx\) \(0.394085 - 0.872510i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.11 + 0.874i)T \)
3 \( 1 \)
7 \( 1 + (1.48 + 2.18i)T \)
good5 \( 1 + (-1.51 + 2.62i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-4.18 + 2.41i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 1.60iT - 13T^{2} \)
17 \( 1 + (5.79 - 3.34i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.663 - 1.14i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.32 + 7.49i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 7.28T + 29T^{2} \)
31 \( 1 + (-6.89 + 3.98i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.70 + 1.55i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 5.27iT - 41T^{2} \)
43 \( 1 + 3.12T + 43T^{2} \)
47 \( 1 + (-0.262 + 0.454i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.00 + 8.66i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.73 - 1.00i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-8.83 - 5.10i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.979 - 1.69i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 2.66T + 71T^{2} \)
73 \( 1 + (1.96 + 3.40i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.66 - 1.54i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 8.22iT - 83T^{2} \)
89 \( 1 + (-9.40 - 5.42i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 15.3T + 97T^{2} \)
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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

−10.54561945904114466801830127149, −9.559035952561602027420517290751, −8.927985609129349110950568387833, −8.386755617648481854506380992437, −6.89358882837686261927215571348, −6.24145154307389420031273958264, −4.51795397035936534910163386386, −3.72248156250205423307325317425, −1.99744735591913672595179630442, −0.75079201009204129116088664746, 1.88737809104981008765477935867, 3.05033679478305149625372332205, 4.93152897639579249682185751885, 6.11314528549381584784014556048, 6.70003549279896771543012461401, 7.35949100353435045682200962001, 8.856265797831029547172101603978, 9.409362054143068073159776614066, 10.04840598636215927844803872850, 11.13166856573423447682139001836

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