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Properties

Label 2-460-115.102-c1-0-9
Degree $2$
Conductor $460$
Sign $0.752 + 0.658i$
Analytic cond. $3.67311$
Root an. cond. $1.91653$
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  + (3.30 − 0.718i)3-s + (−1.74 − 1.39i)5-s + (0.976 − 0.533i)7-s + (7.66 − 3.50i)9-s + (1.18 + 1.03i)11-s + (−2.81 + 5.14i)13-s + (−6.77 − 3.36i)15-s + (−2.58 − 1.93i)17-s + (0.413 − 2.87i)19-s + (2.84 − 2.46i)21-s + (3.78 − 2.95i)23-s + (1.09 + 4.87i)25-s + (14.7 − 11.0i)27-s + (−5.38 + 0.773i)29-s + (−1.93 + 1.24i)31-s + ⋯
L(s)  = 1  + (1.90 − 0.414i)3-s + (−0.780 − 0.624i)5-s + (0.369 − 0.201i)7-s + (2.55 − 1.16i)9-s + (0.358 + 0.310i)11-s + (−0.779 + 1.42i)13-s + (−1.74 − 0.867i)15-s + (−0.625 − 0.468i)17-s + (0.0947 − 0.659i)19-s + (0.620 − 0.537i)21-s + (0.788 − 0.615i)23-s + (0.219 + 0.975i)25-s + (2.82 − 2.11i)27-s + (−0.999 + 0.143i)29-s + (−0.347 + 0.223i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.752 + 0.658i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.752 + 0.658i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $0.752 + 0.658i$
Analytic conductor: \(3.67311\)
Root analytic conductor: \(1.91653\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (217, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 460,\ (\ :1/2),\ 0.752 + 0.658i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.22246 - 0.834797i\)
\(L(\frac12)\) \(\approx\) \(2.22246 - 0.834797i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (1.74 + 1.39i)T \)
23 \( 1 + (-3.78 + 2.95i)T \)
good3 \( 1 + (-3.30 + 0.718i)T + (2.72 - 1.24i)T^{2} \)
7 \( 1 + (-0.976 + 0.533i)T + (3.78 - 5.88i)T^{2} \)
11 \( 1 + (-1.18 - 1.03i)T + (1.56 + 10.8i)T^{2} \)
13 \( 1 + (2.81 - 5.14i)T + (-7.02 - 10.9i)T^{2} \)
17 \( 1 + (2.58 + 1.93i)T + (4.78 + 16.3i)T^{2} \)
19 \( 1 + (-0.413 + 2.87i)T + (-18.2 - 5.35i)T^{2} \)
29 \( 1 + (5.38 - 0.773i)T + (27.8 - 8.17i)T^{2} \)
31 \( 1 + (1.93 - 1.24i)T + (12.8 - 28.1i)T^{2} \)
37 \( 1 + (6.69 + 2.49i)T + (27.9 + 24.2i)T^{2} \)
41 \( 1 + (-0.0672 + 0.147i)T + (-26.8 - 30.9i)T^{2} \)
43 \( 1 + (-1.75 - 8.05i)T + (-39.1 + 17.8i)T^{2} \)
47 \( 1 + (-5.06 - 5.06i)T + 47iT^{2} \)
53 \( 1 + (-1.28 - 2.36i)T + (-28.6 + 44.5i)T^{2} \)
59 \( 1 + (0.348 + 1.18i)T + (-49.6 + 31.8i)T^{2} \)
61 \( 1 + (-5.81 - 9.04i)T + (-25.3 + 55.4i)T^{2} \)
67 \( 1 + (14.1 + 1.00i)T + (66.3 + 9.53i)T^{2} \)
71 \( 1 + (0.836 + 0.965i)T + (-10.1 + 70.2i)T^{2} \)
73 \( 1 + (-9.87 - 13.1i)T + (-20.5 + 70.0i)T^{2} \)
79 \( 1 + (-2.32 + 0.683i)T + (66.4 - 42.7i)T^{2} \)
83 \( 1 + (-1.11 + 2.98i)T + (-62.7 - 54.3i)T^{2} \)
89 \( 1 + (7.38 + 4.74i)T + (36.9 + 80.9i)T^{2} \)
97 \( 1 + (1.61 + 4.32i)T + (-73.3 + 63.5i)T^{2} \)
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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

−11.02427593884137925899283048379, −9.427365340206771076982045949054, −9.148065580503609014665976828672, −8.350926144255222985971796011870, −7.23677849784045132539598944781, −7.04207042265331845837165140757, −4.64326487501652729881518785278, −4.09055010338308667715635299787, −2.75174794603584259592710063687, −1.54783948933203381353292542308, 2.11767123684858120766915052909, 3.26756153598254461239782252110, 3.86060139245965274458748597590, 5.18883174692463791201505207683, 7.01889296138549179401962322835, 7.75707575950862349378647872615, 8.368856429323185615822994478190, 9.202533321110796372851368526043, 10.21286741387501841348694401307, 10.84851725635408792446365197267

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