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Properties

Label 2-460-115.104-c1-0-7
Degree $2$
Conductor $460$
Sign $0.968 - 0.249i$
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  + (2.25 + 0.324i)3-s + (0.569 + 2.16i)5-s + (3.55 − 3.08i)7-s + (2.10 + 0.618i)9-s + (1.12 + 0.725i)11-s + (−3.67 − 3.18i)13-s + (0.582 + 5.06i)15-s + (−5.55 + 2.53i)17-s + (1.84 − 4.04i)19-s + (9.02 − 5.80i)21-s + (0.167 + 4.79i)23-s + (−4.35 + 2.46i)25-s + (−1.67 − 0.763i)27-s + (1.39 + 3.04i)29-s + (1.02 + 7.11i)31-s + ⋯
L(s)  = 1  + (1.30 + 0.187i)3-s + (0.254 + 0.967i)5-s + (1.34 − 1.16i)7-s + (0.701 + 0.206i)9-s + (0.340 + 0.218i)11-s + (−1.01 − 0.882i)13-s + (0.150 + 1.30i)15-s + (−1.34 + 0.615i)17-s + (0.423 − 0.928i)19-s + (1.97 − 1.26i)21-s + (0.0349 + 0.999i)23-s + (−0.870 + 0.492i)25-s + (−0.321 − 0.146i)27-s + (0.258 + 0.565i)29-s + (0.183 + 1.27i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.249i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 - 0.249i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $0.968 - 0.249i$
Analytic conductor: \(3.67311\)
Root analytic conductor: \(1.91653\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 460,\ (\ :1/2),\ 0.968 - 0.249i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.31481 + 0.292995i\)
\(L(\frac12)\) \(\approx\) \(2.31481 + 0.292995i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.569 - 2.16i)T \)
23 \( 1 + (-0.167 - 4.79i)T \)
good3 \( 1 + (-2.25 - 0.324i)T + (2.87 + 0.845i)T^{2} \)
7 \( 1 + (-3.55 + 3.08i)T + (0.996 - 6.92i)T^{2} \)
11 \( 1 + (-1.12 - 0.725i)T + (4.56 + 10.0i)T^{2} \)
13 \( 1 + (3.67 + 3.18i)T + (1.85 + 12.8i)T^{2} \)
17 \( 1 + (5.55 - 2.53i)T + (11.1 - 12.8i)T^{2} \)
19 \( 1 + (-1.84 + 4.04i)T + (-12.4 - 14.3i)T^{2} \)
29 \( 1 + (-1.39 - 3.04i)T + (-18.9 + 21.9i)T^{2} \)
31 \( 1 + (-1.02 - 7.11i)T + (-29.7 + 8.73i)T^{2} \)
37 \( 1 + (-0.942 + 3.20i)T + (-31.1 - 20.0i)T^{2} \)
41 \( 1 + (-1.71 + 0.502i)T + (34.4 - 22.1i)T^{2} \)
43 \( 1 + (5.67 + 0.815i)T + (41.2 + 12.1i)T^{2} \)
47 \( 1 + 11.3iT - 47T^{2} \)
53 \( 1 + (5.72 - 4.95i)T + (7.54 - 52.4i)T^{2} \)
59 \( 1 + (2.17 - 2.50i)T + (-8.39 - 58.3i)T^{2} \)
61 \( 1 + (-1.79 - 12.4i)T + (-58.5 + 17.1i)T^{2} \)
67 \( 1 + (2.75 + 4.28i)T + (-27.8 + 60.9i)T^{2} \)
71 \( 1 + (1.25 - 0.804i)T + (29.4 - 64.5i)T^{2} \)
73 \( 1 + (-12.6 - 5.76i)T + (47.8 + 55.1i)T^{2} \)
79 \( 1 + (-3.76 + 4.34i)T + (-11.2 - 78.1i)T^{2} \)
83 \( 1 + (-0.233 + 0.795i)T + (-69.8 - 44.8i)T^{2} \)
89 \( 1 + (-0.786 + 5.47i)T + (-85.3 - 25.0i)T^{2} \)
97 \( 1 + (2.67 + 9.11i)T + (-81.6 + 52.4i)T^{2} \)
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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

−10.84255456204700964887692132222, −10.27860576150589797208978755299, −9.260528141257520973304161647799, −8.313282611714519552374271783936, −7.45317031312220011718215582011, −6.91412160899564769178729015251, −5.14681336566627621696945862416, −4.06838988875703277892272029628, −2.99957169606544240261663422850, −1.85587715829447054651321506773, 1.83950101359756546325339420566, 2.49968235259028354329099788457, 4.33380868474708221173354499372, 5.04824865851141178945579405664, 6.36511286545729957777768213806, 7.919903038015670484033579596734, 8.250599195390066146187592316923, 9.226736384478911191543697514414, 9.511918963018817830025599662520, 11.29549344224017993801525017863

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