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Properties

Label 2-468-117.94-c1-0-11
Degree $2$
Conductor $468$
Sign $0.201 + 0.979i$
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  + (0.964 − 1.43i)3-s + (−0.468 + 0.812i)5-s − 0.897·7-s + (−1.13 − 2.77i)9-s + (1.14 − 1.98i)11-s + (2.12 − 2.91i)13-s + (0.715 + 1.45i)15-s + (2.42 − 4.19i)17-s + (3.21 − 5.56i)19-s + (−0.865 + 1.29i)21-s − 2.45·23-s + (2.06 + 3.56i)25-s + (−5.09 − 1.04i)27-s + (−5.24 + 9.07i)29-s + (3.53 − 6.12i)31-s + ⋯
L(s)  = 1  + (0.557 − 0.830i)3-s + (−0.209 + 0.363i)5-s − 0.339·7-s + (−0.379 − 0.925i)9-s + (0.345 − 0.597i)11-s + (0.588 − 0.808i)13-s + (0.184 + 0.376i)15-s + (0.587 − 1.01i)17-s + (0.737 − 1.27i)19-s + (−0.188 + 0.281i)21-s − 0.512·23-s + (0.412 + 0.713i)25-s + (−0.979 − 0.200i)27-s + (−0.973 + 1.68i)29-s + (0.635 − 1.10i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 468 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.201 + 0.979i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 468 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.201 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(468\)    =    \(2^{2} \cdot 3^{2} \cdot 13\)
Sign: $0.201 + 0.979i$
Analytic conductor: \(3.73699\)
Root analytic conductor: \(1.93313\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{468} (445, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 468,\ (\ :1/2),\ 0.201 + 0.979i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.20946 - 0.986402i\)
\(L(\frac12)\) \(\approx\) \(1.20946 - 0.986402i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.964 + 1.43i)T \)
13 \( 1 + (-2.12 + 2.91i)T \)
good5 \( 1 + (0.468 - 0.812i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + 0.897T + 7T^{2} \)
11 \( 1 + (-1.14 + 1.98i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-2.42 + 4.19i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.21 + 5.56i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 2.45T + 23T^{2} \)
29 \( 1 + (5.24 - 9.07i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.53 + 6.12i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.82 - 4.90i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 1.58T + 41T^{2} \)
43 \( 1 + 4.11T + 43T^{2} \)
47 \( 1 + (-4.34 - 7.52i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 9.76T + 53T^{2} \)
59 \( 1 + (-6.06 - 10.4i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 - 7.42T + 61T^{2} \)
67 \( 1 - 10.8T + 67T^{2} \)
71 \( 1 + (-2.14 + 3.72i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 6.91T + 73T^{2} \)
79 \( 1 + (3.32 + 5.75i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.852 - 1.47i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-0.452 - 0.783i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 4.65T + 97T^{2} \)
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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

−11.10881439596025808378435389300, −9.728045990604393137131058354102, −9.006812017817925409137432877880, −7.999616929960024263321274784491, −7.22080600568721302972364692126, −6.37849764984533097942647381391, −5.27558696498360334164748784650, −3.48267075463470155759313913635, −2.85692729109594977811499619136, −0.984091100354097237102718053150, 1.90076067574012782541801458897, 3.59012684444120737338107208796, 4.18131793494767353935993233589, 5.44279499400242662442920785208, 6.53602198544048401024887884214, 7.939507340413311936688155263417, 8.477040460258494382373227592381, 9.677467704030066587681554125777, 9.979583589186005923834501281284, 11.16273848903285327944119575638

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