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Properties

Label 2-468-117.103-c1-0-1
Degree $2$
Conductor $468$
Sign $-0.415 - 0.909i$
Analytic cond. $3.73699$
Root an. cond. $1.93313$
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  + (−1.73 + 0.0398i)3-s + (−3.64 − 2.10i)5-s + (2.60 − 1.50i)7-s + (2.99 − 0.138i)9-s + (−1.65 + 0.953i)11-s + (−1.12 + 3.42i)13-s + (6.39 + 3.49i)15-s − 3.93·17-s + 1.37i·19-s + (−4.45 + 2.71i)21-s + (−2.04 + 3.54i)23-s + (6.36 + 11.0i)25-s + (−5.18 + 0.358i)27-s + (4.04 + 7.01i)29-s + (−8.05 − 4.65i)31-s + ⋯
L(s)  = 1  + (−0.999 + 0.0230i)3-s + (−1.63 − 0.941i)5-s + (0.985 − 0.569i)7-s + (0.998 − 0.0460i)9-s + (−0.497 + 0.287i)11-s + (−0.310 + 0.950i)13-s + (1.65 + 0.903i)15-s − 0.953·17-s + 0.315i·19-s + (−0.972 + 0.591i)21-s + (−0.426 + 0.739i)23-s + (1.27 + 2.20i)25-s + (−0.997 + 0.0690i)27-s + (0.752 + 1.30i)29-s + (−1.44 − 0.835i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 468 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.415 - 0.909i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 468 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.415 - 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(468\)    =    \(2^{2} \cdot 3^{2} \cdot 13\)
Sign: $-0.415 - 0.909i$
Analytic conductor: \(3.73699\)
Root analytic conductor: \(1.93313\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{468} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 468,\ (\ :1/2),\ -0.415 - 0.909i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.137578 + 0.214015i\)
\(L(\frac12)\) \(\approx\) \(0.137578 + 0.214015i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.73 - 0.0398i)T \)
13 \( 1 + (1.12 - 3.42i)T \)
good5 \( 1 + (3.64 + 2.10i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (-2.60 + 1.50i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.65 - 0.953i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + 3.93T + 17T^{2} \)
19 \( 1 - 1.37iT - 19T^{2} \)
23 \( 1 + (2.04 - 3.54i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.04 - 7.01i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (8.05 + 4.65i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 4.12iT - 37T^{2} \)
41 \( 1 + (-3.14 - 1.81i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.278 - 0.482i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.89 + 2.24i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 - 0.138T + 53T^{2} \)
59 \( 1 + (-2.78 - 1.60i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.34 + 4.06i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (8.84 + 5.10i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 6.13iT - 71T^{2} \)
73 \( 1 - 11.4iT - 73T^{2} \)
79 \( 1 + (-2.66 - 4.60i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (11.6 - 6.72i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 13.5iT - 89T^{2} \)
97 \( 1 + (12.1 - 7.00i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.22038371502179221567168767640, −10.94343602887132920595714551494, −9.528428511675673323722809505556, −8.440065137134219413530569042447, −7.58001712603113808620434687096, −6.96890251283103934417389418864, −5.32978658383604974310846628099, −4.53594435720283526398933038638, −4.00277120472402022235769809176, −1.44998957575625054905241456800, 0.18704056065806499085198249930, 2.57654197592242314769853573873, 4.05046414179641897923010421859, 4.92847465919522540089125261182, 6.05513208076511261254010204104, 7.19815943022925460679484664479, 7.81115426683559229352536769963, 8.698284444972060794665503199364, 10.42923318888154837143260278443, 10.83969751058718865539559707373

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