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Properties

Label 2-460-115.102-c1-0-3
Degree $2$
Conductor $460$
Sign $0.964 + 0.262i$
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  + (−2.90 + 0.632i)3-s + (−2.22 − 0.170i)5-s + (−1.58 + 0.867i)7-s + (5.31 − 2.42i)9-s + (−2.69 − 2.33i)11-s + (−1.24 + 2.27i)13-s + (6.58 − 0.913i)15-s + (3.95 + 2.96i)17-s + (0.433 − 3.01i)19-s + (4.06 − 3.52i)21-s + (4.16 − 2.37i)23-s + (4.94 + 0.760i)25-s + (−6.76 + 5.06i)27-s + (7.93 − 1.14i)29-s + (−2.32 + 1.49i)31-s + ⋯
L(s)  = 1  + (−1.67 + 0.364i)3-s + (−0.997 − 0.0762i)5-s + (−0.600 + 0.327i)7-s + (1.77 − 0.809i)9-s + (−0.812 − 0.704i)11-s + (−0.344 + 0.631i)13-s + (1.70 − 0.235i)15-s + (0.959 + 0.718i)17-s + (0.0993 − 0.691i)19-s + (0.887 − 0.769i)21-s + (0.869 − 0.494i)23-s + (0.988 + 0.152i)25-s + (−1.30 + 0.974i)27-s + (1.47 − 0.211i)29-s + (−0.416 + 0.267i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 + 0.262i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.964 + 0.262i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $0.964 + 0.262i$
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: \((2,\ 460,\ (\ :1/2),\ 0.964 + 0.262i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.472196 - 0.0631383i\)
\(L(\frac12)\) \(\approx\) \(0.472196 - 0.0631383i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.22 + 0.170i)T \)
23 \( 1 + (-4.16 + 2.37i)T \)
good3 \( 1 + (2.90 - 0.632i)T + (2.72 - 1.24i)T^{2} \)
7 \( 1 + (1.58 - 0.867i)T + (3.78 - 5.88i)T^{2} \)
11 \( 1 + (2.69 + 2.33i)T + (1.56 + 10.8i)T^{2} \)
13 \( 1 + (1.24 - 2.27i)T + (-7.02 - 10.9i)T^{2} \)
17 \( 1 + (-3.95 - 2.96i)T + (4.78 + 16.3i)T^{2} \)
19 \( 1 + (-0.433 + 3.01i)T + (-18.2 - 5.35i)T^{2} \)
29 \( 1 + (-7.93 + 1.14i)T + (27.8 - 8.17i)T^{2} \)
31 \( 1 + (2.32 - 1.49i)T + (12.8 - 28.1i)T^{2} \)
37 \( 1 + (-0.715 - 0.266i)T + (27.9 + 24.2i)T^{2} \)
41 \( 1 + (-4.20 + 9.21i)T + (-26.8 - 30.9i)T^{2} \)
43 \( 1 + (-1.08 - 4.98i)T + (-39.1 + 17.8i)T^{2} \)
47 \( 1 + (3.62 + 3.62i)T + 47iT^{2} \)
53 \( 1 + (5.10 + 9.35i)T + (-28.6 + 44.5i)T^{2} \)
59 \( 1 + (-4.20 - 14.3i)T + (-49.6 + 31.8i)T^{2} \)
61 \( 1 + (5.27 + 8.21i)T + (-25.3 + 55.4i)T^{2} \)
67 \( 1 + (-2.15 - 0.154i)T + (66.3 + 9.53i)T^{2} \)
71 \( 1 + (0.739 + 0.853i)T + (-10.1 + 70.2i)T^{2} \)
73 \( 1 + (1.48 + 1.98i)T + (-20.5 + 70.0i)T^{2} \)
79 \( 1 + (-8.36 + 2.45i)T + (66.4 - 42.7i)T^{2} \)
83 \( 1 + (-2.94 + 7.90i)T + (-62.7 - 54.3i)T^{2} \)
89 \( 1 + (-12.6 - 8.10i)T + (36.9 + 80.9i)T^{2} \)
97 \( 1 + (-0.462 - 1.24i)T + (-73.3 + 63.5i)T^{2} \)
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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

−10.95755376279298100533945959351, −10.52609152469811312093137329924, −9.404687078626960174082349435158, −8.256425088047562790894102395931, −7.08686728486536430804169575297, −6.26105700338756835459401468842, −5.27685124214768985359265009319, −4.47566265605935781932108302298, −3.17921738538068129378573614021, −0.57077781999024799243739544307, 0.821942598345817773852935773855, 3.14371403229235225134430295761, 4.63941670672358934551546328161, 5.34449453403728969057184092497, 6.48220073984713359944904906918, 7.35862426813146436745378985860, 7.899394907005045905378093610674, 9.700122458496490204033526968300, 10.41802930233611235436243722249, 11.14853110165342670881258220560

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