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Properties

Label 2-468-117.103-c1-0-12
Degree $2$
Conductor $468$
Sign $0.271 + 0.962i$
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  + (1.61 − 0.613i)3-s + (−2.41 − 1.39i)5-s + (1.00 − 0.582i)7-s + (2.24 − 1.98i)9-s + (−0.716 + 0.413i)11-s + (3.28 − 1.48i)13-s + (−4.76 − 0.775i)15-s + 0.0570·17-s − 8.44i·19-s + (1.27 − 1.56i)21-s + (−1.18 + 2.04i)23-s + (1.37 + 2.38i)25-s + (2.42 − 4.59i)27-s + (−2.67 − 4.62i)29-s + (−0.351 − 0.203i)31-s + ⋯
L(s)  = 1  + (0.935 − 0.354i)3-s + (−1.07 − 0.622i)5-s + (0.381 − 0.219i)7-s + (0.749 − 0.662i)9-s + (−0.216 + 0.124i)11-s + (0.911 − 0.410i)13-s + (−1.22 − 0.200i)15-s + 0.0138·17-s − 1.93i·19-s + (0.278 − 0.340i)21-s + (−0.246 + 0.426i)23-s + (0.275 + 0.477i)25-s + (0.465 − 0.884i)27-s + (−0.495 − 0.858i)29-s + (−0.0631 − 0.0364i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 468 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.271 + 0.962i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 468 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.271 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(468\)    =    \(2^{2} \cdot 3^{2} \cdot 13\)
Sign: $0.271 + 0.962i$
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: \((2,\ 468,\ (\ :1/2),\ 0.271 + 0.962i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.31557 - 0.995701i\)
\(L(\frac12)\) \(\approx\) \(1.31557 - 0.995701i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.61 + 0.613i)T \)
13 \( 1 + (-3.28 + 1.48i)T \)
good5 \( 1 + (2.41 + 1.39i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (-1.00 + 0.582i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.716 - 0.413i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 - 0.0570T + 17T^{2} \)
19 \( 1 + 8.44iT - 19T^{2} \)
23 \( 1 + (1.18 - 2.04i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.67 + 4.62i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.351 + 0.203i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 5.42iT - 37T^{2} \)
41 \( 1 + (-7.53 - 4.35i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.13 - 8.90i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (7.56 - 4.36i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 2.12T + 53T^{2} \)
59 \( 1 + (-9.68 - 5.58i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.225 + 0.389i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-8.02 - 4.63i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 6.15iT - 71T^{2} \)
73 \( 1 + 8.45iT - 73T^{2} \)
79 \( 1 + (-4.40 - 7.62i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-9.12 + 5.26i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 5.24iT - 89T^{2} \)
97 \( 1 + (-0.0761 + 0.0439i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.09830735237538102955856086365, −9.705969215527843994521208797281, −8.866574906917639714876021830099, −8.026379287087540490440730726684, −7.58353372702064739059565751018, −6.37448181013669015023185277901, −4.77535084600592584502737138160, −3.96436242549044687630698719879, −2.76055716314264569487038322677, −1.00782772544647225996645370186, 2.00106643257047654431058094391, 3.55104051566561665643872999895, 3.95213852968429091169480720570, 5.45598866657797814900274447212, 6.84838400767920279290385084691, 7.85092057636697169115999128855, 8.325640002616670212193321867543, 9.306620949759630459868544819698, 10.46973686505724767314265117421, 11.02353945558056040554982314402

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