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Properties

Label 2-469-469.328-c0-0-0
Degree $2$
Conductor $469$
Sign $0.733 + 0.679i$
Analytic cond. $0.234061$
Root an. cond. $0.483799$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.16 − 0.600i)2-s + (0.414 − 0.582i)4-s + (0.0475 − 0.998i)7-s + (−0.0530 + 0.368i)8-s + (0.841 + 0.540i)9-s + (−1.74 − 0.336i)11-s + (−0.544 − 1.19i)14-s + (0.393 + 1.13i)16-s + (1.30 + 0.124i)18-s + (−2.23 + 0.655i)22-s + (−1.13 + 1.08i)23-s + (−0.142 − 0.989i)25-s + (−0.562 − 0.442i)28-s + (−0.0475 − 0.0824i)29-s + (0.871 + 0.830i)32-s + ⋯
L(s)  = 1  + (1.16 − 0.600i)2-s + (0.414 − 0.582i)4-s + (0.0475 − 0.998i)7-s + (−0.0530 + 0.368i)8-s + (0.841 + 0.540i)9-s + (−1.74 − 0.336i)11-s + (−0.544 − 1.19i)14-s + (0.393 + 1.13i)16-s + (1.30 + 0.124i)18-s + (−2.23 + 0.655i)22-s + (−1.13 + 1.08i)23-s + (−0.142 − 0.989i)25-s + (−0.562 − 0.442i)28-s + (−0.0475 − 0.0824i)29-s + (0.871 + 0.830i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 469 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.733 + 0.679i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 469 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.733 + 0.679i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(469\)    =    \(7 \cdot 67\)
Sign: $0.733 + 0.679i$
Analytic conductor: \(0.234061\)
Root analytic conductor: \(0.483799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{469} (328, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 469,\ (\ :0),\ 0.733 + 0.679i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.433444626\)
\(L(\frac12)\) \(\approx\) \(1.433444626\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.0475 + 0.998i)T \)
67 \( 1 + (0.327 + 0.945i)T \)
good2 \( 1 + (-1.16 + 0.600i)T + (0.580 - 0.814i)T^{2} \)
3 \( 1 + (-0.841 - 0.540i)T^{2} \)
5 \( 1 + (0.142 + 0.989i)T^{2} \)
11 \( 1 + (1.74 + 0.336i)T + (0.928 + 0.371i)T^{2} \)
13 \( 1 + (-0.723 - 0.690i)T^{2} \)
17 \( 1 + (0.327 - 0.945i)T^{2} \)
19 \( 1 + (0.995 - 0.0950i)T^{2} \)
23 \( 1 + (1.13 - 1.08i)T + (0.0475 - 0.998i)T^{2} \)
29 \( 1 + (0.0475 + 0.0824i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (-0.723 + 0.690i)T^{2} \)
37 \( 1 + (-0.327 + 0.566i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.981 + 0.189i)T^{2} \)
43 \( 1 + (0.827 - 1.81i)T + (-0.654 - 0.755i)T^{2} \)
47 \( 1 + (0.888 - 0.458i)T^{2} \)
53 \( 1 + (-0.195 - 0.428i)T + (-0.654 + 0.755i)T^{2} \)
59 \( 1 + (0.959 + 0.281i)T^{2} \)
61 \( 1 + (-0.928 + 0.371i)T^{2} \)
71 \( 1 + (-1.13 + 1.59i)T + (-0.327 - 0.945i)T^{2} \)
73 \( 1 + (-0.928 + 0.371i)T^{2} \)
79 \( 1 + (-1.50 + 1.18i)T + (0.235 - 0.971i)T^{2} \)
83 \( 1 + (0.786 - 0.618i)T^{2} \)
89 \( 1 + (-0.841 + 0.540i)T^{2} \)
97 \( 1 + (0.5 + 0.866i)T^{2} \)
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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

−11.10519953537118682533023454566, −10.54611055624268556914350969300, −9.800307408881554045781843115291, −8.033176138825284140839618282525, −7.67746265334078587352711503208, −6.18413913674320507387247757004, −5.07986043777179552831616419702, −4.36506826404411696555754100051, −3.30158705569475361653790373230, −2.03358300396673131607422109053, 2.39695158864687070922694626565, 3.71948541118468979371284338538, 4.88833022412559096761458735561, 5.53340089432031294787321498267, 6.51897647948453703747034975981, 7.44718259867941232793780483929, 8.454711681690250623389281512595, 9.704216738971640550295997949765, 10.34788868122198046768469135322, 11.72983407544884258367938203118

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