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Properties

Label 2-468-117.61-c1-0-7
Degree $2$
Conductor $468$
Sign $0.491 - 0.871i$
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  + (1.36 + 1.07i)3-s + (1.62 + 2.81i)5-s + 4.02·7-s + (0.700 + 2.91i)9-s + (−2.72 − 4.71i)11-s + (−1.08 + 3.43i)13-s + (−0.807 + 5.56i)15-s + (−3.51 − 6.08i)17-s + (−3.02 − 5.24i)19-s + (5.47 + 4.31i)21-s + 0.705·23-s + (−2.77 + 4.81i)25-s + (−2.17 + 4.71i)27-s + (−1.32 − 2.29i)29-s + (2.87 + 4.97i)31-s + ⋯
L(s)  = 1  + (0.785 + 0.619i)3-s + (0.726 + 1.25i)5-s + 1.52·7-s + (0.233 + 0.972i)9-s + (−0.820 − 1.42i)11-s + (−0.301 + 0.953i)13-s + (−0.208 + 1.43i)15-s + (−0.852 − 1.47i)17-s + (−0.694 − 1.20i)19-s + (1.19 + 0.942i)21-s + 0.147·23-s + (−0.555 + 0.962i)25-s + (−0.418 + 0.908i)27-s + (−0.246 − 0.426i)29-s + (0.516 + 0.894i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 468 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.491 - 0.871i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 468 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.491 - 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(468\)    =    \(2^{2} \cdot 3^{2} \cdot 13\)
Sign: $0.491 - 0.871i$
Analytic conductor: \(3.73699\)
Root analytic conductor: \(1.93313\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{468} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 468,\ (\ :1/2),\ 0.491 - 0.871i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.86826 + 1.09128i\)
\(L(\frac12)\) \(\approx\) \(1.86826 + 1.09128i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.36 - 1.07i)T \)
13 \( 1 + (1.08 - 3.43i)T \)
good5 \( 1 + (-1.62 - 2.81i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 - 4.02T + 7T^{2} \)
11 \( 1 + (2.72 + 4.71i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (3.51 + 6.08i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.02 + 5.24i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 0.705T + 23T^{2} \)
29 \( 1 + (1.32 + 2.29i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.87 - 4.97i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.965 + 1.67i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 4.07T + 41T^{2} \)
43 \( 1 - 5.42T + 43T^{2} \)
47 \( 1 + (-1.62 + 2.80i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 0.0982T + 53T^{2} \)
59 \( 1 + (3.47 - 6.02i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + 7.95T + 61T^{2} \)
67 \( 1 + 0.0908T + 67T^{2} \)
71 \( 1 + (-2.62 - 4.54i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 4.40T + 73T^{2} \)
79 \( 1 + (-2.93 + 5.08i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.26 - 5.65i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (3.12 - 5.41i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 16.7T + 97T^{2} \)
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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

−10.99467711593635326458754097823, −10.50821919870418121482039778873, −9.276629920130030474662607656392, −8.630692462323486975267626798331, −7.59496515898491495825630048748, −6.69441815694045913275303274074, −5.30062825009349243714039068558, −4.46653287832115990494525793173, −2.87831721575217371575689829668, −2.24082407844397375478867295371, 1.55801896842510269002287894623, 2.17242960316136936181126021752, 4.23788334160059517051262195533, 5.05142926876561519484696859241, 6.12601269237866593766649781490, 7.68205684287475337163628468431, 8.096584800703361582586209737483, 8.853478877877623514383608873406, 9.900133618305666285458528340076, 10.71506218610949793923477076002

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