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Properties

Label 2-468-117.61-c1-0-11
Degree $2$
Conductor $468$
Sign $-0.551 + 0.834i$
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  + (0.583 − 1.63i)3-s + (−1.31 − 2.27i)5-s + 3.59·7-s + (−2.31 − 1.90i)9-s + (−2.37 − 4.11i)11-s + (−3.08 + 1.87i)13-s + (−4.47 + 0.813i)15-s + (0.733 + 1.27i)17-s + (2.78 + 4.82i)19-s + (2.09 − 5.85i)21-s + 3.41·23-s + (−0.942 + 1.63i)25-s + (−4.45 + 2.66i)27-s + (−1.91 − 3.32i)29-s + (−4.73 − 8.20i)31-s + ⋯
L(s)  = 1  + (0.336 − 0.941i)3-s + (−0.586 − 1.01i)5-s + 1.35·7-s + (−0.772 − 0.634i)9-s + (−0.716 − 1.24i)11-s + (−0.854 + 0.519i)13-s + (−1.15 + 0.209i)15-s + (0.177 + 0.308i)17-s + (0.639 + 1.10i)19-s + (0.457 − 1.27i)21-s + 0.711·23-s + (−0.188 + 0.326i)25-s + (−0.857 + 0.513i)27-s + (−0.356 − 0.617i)29-s + (−0.850 − 1.47i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 468 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.551 + 0.834i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 468 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.551 + 0.834i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(468\)    =    \(2^{2} \cdot 3^{2} \cdot 13\)
Sign: $-0.551 + 0.834i$
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: \((2,\ 468,\ (\ :1/2),\ -0.551 + 0.834i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.658864 - 1.22487i\)
\(L(\frac12)\) \(\approx\) \(0.658864 - 1.22487i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.583 + 1.63i)T \)
13 \( 1 + (3.08 - 1.87i)T \)
good5 \( 1 + (1.31 + 2.27i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 - 3.59T + 7T^{2} \)
11 \( 1 + (2.37 + 4.11i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-0.733 - 1.27i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.78 - 4.82i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 3.41T + 23T^{2} \)
29 \( 1 + (1.91 + 3.32i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.73 + 8.20i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.14 + 3.71i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 7.34T + 41T^{2} \)
43 \( 1 + 2.56T + 43T^{2} \)
47 \( 1 + (3.88 - 6.72i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 1.77T + 53T^{2} \)
59 \( 1 + (-4.39 + 7.61i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 - 6.37T + 61T^{2} \)
67 \( 1 - 4.24T + 67T^{2} \)
71 \( 1 + (-4.76 - 8.25i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 11.1T + 73T^{2} \)
79 \( 1 + (5.06 - 8.77i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.51 - 7.82i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-6.36 + 11.0i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 12.0T + 97T^{2} \)
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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

−11.23393044115097097152678962891, −9.589660194112179952111182937792, −8.566342684199211061495013977796, −7.976662942170808719709059286513, −7.50386952929960349836789935562, −5.89218066117836590522277585419, −5.07496096719836766670401671952, −3.82189424461650206549215241703, −2.21500138421751393530955562503, −0.850038336254408669699253402061, 2.36534842579509694951653473628, 3.35905582766403615311797268895, 4.88455212214592269913145237576, 5.06979987162710466318783328128, 7.20533911809244551103790573617, 7.53449427390448780721486922209, 8.639298563046091521517835374814, 9.702741807337137379612308704994, 10.55815649132971929796167539879, 11.11256873685278709790483578110

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