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Properties

Label 2-460-115.102-c1-0-6
Degree $2$
Conductor $460$
Sign $0.252 + 0.967i$
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  + (−2.23 + 0.486i)3-s + (1.73 − 1.41i)5-s + (−2.25 + 1.23i)7-s + (2.04 − 0.932i)9-s + (0.879 + 0.761i)11-s + (2.27 − 4.16i)13-s + (−3.18 + 4.01i)15-s + (−3.35 − 2.50i)17-s + (0.0362 − 0.252i)19-s + (4.45 − 3.85i)21-s + (−1.31 − 4.61i)23-s + (0.992 − 4.90i)25-s + (1.38 − 1.03i)27-s + (1.34 − 0.194i)29-s + (8.60 − 5.53i)31-s + ⋯
L(s)  = 1  + (−1.29 + 0.281i)3-s + (0.774 − 0.633i)5-s + (−0.853 + 0.466i)7-s + (0.680 − 0.310i)9-s + (0.265 + 0.229i)11-s + (0.631 − 1.15i)13-s + (−0.822 + 1.03i)15-s + (−0.812 − 0.608i)17-s + (0.00831 − 0.0578i)19-s + (0.971 − 0.842i)21-s + (−0.273 − 0.961i)23-s + (0.198 − 0.980i)25-s + (0.266 − 0.199i)27-s + (0.250 − 0.0360i)29-s + (1.54 − 0.993i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.252 + 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.252 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $0.252 + 0.967i$
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: \((2,\ 460,\ (\ :1/2),\ 0.252 + 0.967i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.605838 - 0.467889i\)
\(L(\frac12)\) \(\approx\) \(0.605838 - 0.467889i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.73 + 1.41i)T \)
23 \( 1 + (1.31 + 4.61i)T \)
good3 \( 1 + (2.23 - 0.486i)T + (2.72 - 1.24i)T^{2} \)
7 \( 1 + (2.25 - 1.23i)T + (3.78 - 5.88i)T^{2} \)
11 \( 1 + (-0.879 - 0.761i)T + (1.56 + 10.8i)T^{2} \)
13 \( 1 + (-2.27 + 4.16i)T + (-7.02 - 10.9i)T^{2} \)
17 \( 1 + (3.35 + 2.50i)T + (4.78 + 16.3i)T^{2} \)
19 \( 1 + (-0.0362 + 0.252i)T + (-18.2 - 5.35i)T^{2} \)
29 \( 1 + (-1.34 + 0.194i)T + (27.8 - 8.17i)T^{2} \)
31 \( 1 + (-8.60 + 5.53i)T + (12.8 - 28.1i)T^{2} \)
37 \( 1 + (4.45 + 1.66i)T + (27.9 + 24.2i)T^{2} \)
41 \( 1 + (-2.79 + 6.12i)T + (-26.8 - 30.9i)T^{2} \)
43 \( 1 + (1.36 + 6.28i)T + (-39.1 + 17.8i)T^{2} \)
47 \( 1 + (1.71 + 1.71i)T + 47iT^{2} \)
53 \( 1 + (-3.84 - 7.04i)T + (-28.6 + 44.5i)T^{2} \)
59 \( 1 + (0.664 + 2.26i)T + (-49.6 + 31.8i)T^{2} \)
61 \( 1 + (0.532 + 0.828i)T + (-25.3 + 55.4i)T^{2} \)
67 \( 1 + (8.75 + 0.626i)T + (66.3 + 9.53i)T^{2} \)
71 \( 1 + (4.15 + 4.79i)T + (-10.1 + 70.2i)T^{2} \)
73 \( 1 + (-8.88 - 11.8i)T + (-20.5 + 70.0i)T^{2} \)
79 \( 1 + (0.671 - 0.197i)T + (66.4 - 42.7i)T^{2} \)
83 \( 1 + (4.58 - 12.2i)T + (-62.7 - 54.3i)T^{2} \)
89 \( 1 + (8.57 + 5.51i)T + (36.9 + 80.9i)T^{2} \)
97 \( 1 + (-5.41 - 14.5i)T + (-73.3 + 63.5i)T^{2} \)
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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

−10.72498642727522093424228078238, −10.14519947752796085421618615745, −9.232619744093750476257756555121, −8.349408431051932724945145939724, −6.71209721245815773953707545483, −6.03226847024071913412660307861, −5.35084292735997545700790154774, −4.32743928716550611255649577067, −2.59477100743784851256940643249, −0.58193567986174741379418589674, 1.49749195787092955659592614751, 3.27835096154582850102676124805, 4.63825634718043399677106952022, 6.01727989955647935946372854363, 6.42054907696215041998235113773, 7.04666637220884887960728234269, 8.666368323766467678565009851909, 9.728729162063889331979038843730, 10.45597144150181624046941386598, 11.27283451077801371689489051713

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