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Properties

Label 2-468-117.103-c1-0-10
Degree $2$
Conductor $468$
Sign $-0.0245 + 0.999i$
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  + (−1.05 − 1.37i)3-s + (0.154 + 0.0889i)5-s + (1.97 − 1.13i)7-s + (−0.781 + 2.89i)9-s + (3.52 − 2.03i)11-s + (−3.23 − 1.59i)13-s + (−0.0399 − 0.305i)15-s + 3.07·17-s − 5.12i·19-s + (−3.63 − 1.51i)21-s + (−1.90 + 3.29i)23-s + (−2.48 − 4.30i)25-s + (4.80 − 1.97i)27-s + (−1.54 − 2.68i)29-s + (−2.83 − 1.63i)31-s + ⋯
L(s)  = 1  + (−0.608 − 0.793i)3-s + (0.0688 + 0.0397i)5-s + (0.744 − 0.429i)7-s + (−0.260 + 0.965i)9-s + (1.06 − 0.613i)11-s + (−0.896 − 0.443i)13-s + (−0.0103 − 0.0788i)15-s + 0.746·17-s − 1.17i·19-s + (−0.794 − 0.329i)21-s + (−0.396 + 0.686i)23-s + (−0.496 − 0.860i)25-s + (0.924 − 0.380i)27-s + (−0.287 − 0.498i)29-s + (−0.510 − 0.294i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 468 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0245 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 468 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0245 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(468\)    =    \(2^{2} \cdot 3^{2} \cdot 13\)
Sign: $-0.0245 + 0.999i$
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: \((2,\ 468,\ (\ :1/2),\ -0.0245 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.828180 - 0.848781i\)
\(L(\frac12)\) \(\approx\) \(0.828180 - 0.848781i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.05 + 1.37i)T \)
13 \( 1 + (3.23 + 1.59i)T \)
good5 \( 1 + (-0.154 - 0.0889i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (-1.97 + 1.13i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-3.52 + 2.03i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 - 3.07T + 17T^{2} \)
19 \( 1 + 5.12iT - 19T^{2} \)
23 \( 1 + (1.90 - 3.29i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.54 + 2.68i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.83 + 1.63i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 9.78iT - 37T^{2} \)
41 \( 1 + (0.887 + 0.512i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.98 - 5.17i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-8.53 + 4.92i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 4.76T + 53T^{2} \)
59 \( 1 + (-8.49 - 4.90i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.84 - 4.92i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.80 + 1.04i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 13.5iT - 71T^{2} \)
73 \( 1 - 1.70iT - 73T^{2} \)
79 \( 1 + (1.49 + 2.58i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-8.49 + 4.90i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 16.6iT - 89T^{2} \)
97 \( 1 + (-4.90 + 2.82i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.06414590169451949772281236249, −10.05063810143114149384350374479, −8.930318438223523410409997074279, −7.78313193079064338854638688496, −7.25354849886671485830774026555, −6.09728566545039120101353219000, −5.26212477449628424573226409905, −4.04049747404132805710575167908, −2.32369126202035919963696604421, −0.846561739101029329435707235010, 1.72537758015853580523005072445, 3.58435527133159610505691532087, 4.61643580184386328986915838140, 5.44830973557852998021162492366, 6.47044149964587602659188003841, 7.59528137576282107041346836516, 8.779496244664010814113087756771, 9.610345219594549829176948964460, 10.26067989706161079010332067533, 11.35980334385250597011291932677

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