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Properties

Label 2-460-115.107-c1-0-5
Degree $2$
Conductor $460$
Sign $0.648 - 0.760i$
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  + (2.79 + 1.04i)3-s + (−1.70 + 1.44i)5-s + (3.38 − 0.736i)7-s + (4.44 + 3.84i)9-s + (−1.93 + 0.278i)11-s + (−0.119 + 0.550i)13-s + (−6.27 + 2.25i)15-s + (1.65 − 3.03i)17-s + (−7.17 − 2.10i)19-s + (10.2 + 1.46i)21-s + (3.32 + 3.45i)23-s + (0.829 − 4.93i)25-s + (4.11 + 7.52i)27-s + (2.40 + 8.18i)29-s + (0.0495 − 0.108i)31-s + ⋯
L(s)  = 1  + (1.61 + 0.601i)3-s + (−0.763 + 0.645i)5-s + (1.27 − 0.278i)7-s + (1.48 + 1.28i)9-s + (−0.584 + 0.0839i)11-s + (−0.0332 + 0.152i)13-s + (−1.61 + 0.581i)15-s + (0.401 − 0.736i)17-s + (−1.64 − 0.483i)19-s + (2.22 + 0.320i)21-s + (0.692 + 0.721i)23-s + (0.165 − 0.986i)25-s + (0.791 + 1.44i)27-s + (0.446 + 1.51i)29-s + (0.00889 − 0.0194i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.648 - 0.760i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.648 - 0.760i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $0.648 - 0.760i$
Analytic conductor: \(3.67311\)
Root analytic conductor: \(1.91653\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 460,\ (\ :1/2),\ 0.648 - 0.760i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.05387 + 0.947767i\)
\(L(\frac12)\) \(\approx\) \(2.05387 + 0.947767i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.70 - 1.44i)T \)
23 \( 1 + (-3.32 - 3.45i)T \)
good3 \( 1 + (-2.79 - 1.04i)T + (2.26 + 1.96i)T^{2} \)
7 \( 1 + (-3.38 + 0.736i)T + (6.36 - 2.90i)T^{2} \)
11 \( 1 + (1.93 - 0.278i)T + (10.5 - 3.09i)T^{2} \)
13 \( 1 + (0.119 - 0.550i)T + (-11.8 - 5.40i)T^{2} \)
17 \( 1 + (-1.65 + 3.03i)T + (-9.19 - 14.3i)T^{2} \)
19 \( 1 + (7.17 + 2.10i)T + (15.9 + 10.2i)T^{2} \)
29 \( 1 + (-2.40 - 8.18i)T + (-24.3 + 15.6i)T^{2} \)
31 \( 1 + (-0.0495 + 0.108i)T + (-20.3 - 23.4i)T^{2} \)
37 \( 1 + (0.578 + 8.08i)T + (-36.6 + 5.26i)T^{2} \)
41 \( 1 + (3.79 + 4.37i)T + (-5.83 + 40.5i)T^{2} \)
43 \( 1 + (-2.55 + 6.86i)T + (-32.4 - 28.1i)T^{2} \)
47 \( 1 + (5.07 + 5.07i)T + 47iT^{2} \)
53 \( 1 + (-2.40 - 11.0i)T + (-48.2 + 22.0i)T^{2} \)
59 \( 1 + (-2.80 - 4.36i)T + (-24.5 + 53.6i)T^{2} \)
61 \( 1 + (3.40 + 1.55i)T + (39.9 + 46.1i)T^{2} \)
67 \( 1 + (3.16 + 4.22i)T + (-18.8 + 64.2i)T^{2} \)
71 \( 1 + (-0.382 + 2.66i)T + (-68.1 - 20.0i)T^{2} \)
73 \( 1 + (0.782 - 0.427i)T + (39.4 - 61.4i)T^{2} \)
79 \( 1 + (-10.2 + 6.58i)T + (32.8 - 71.8i)T^{2} \)
83 \( 1 + (1.50 - 0.107i)T + (82.1 - 11.8i)T^{2} \)
89 \( 1 + (7.53 + 16.4i)T + (-58.2 + 67.2i)T^{2} \)
97 \( 1 + (-14.2 - 1.01i)T + (96.0 + 13.8i)T^{2} \)
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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

−10.74637779208808524734162394491, −10.54268980436735934134159222643, −9.096894246034366602810285439427, −8.519965655508712591540506046408, −7.63987977886161382697783901260, −7.10687359617180744914835128461, −5.03552916977547840870669992518, −4.19524581205572251832684263567, −3.18871568830264767379096249856, −2.11230705476185559778202217739, 1.48873088239412114156282946199, 2.64742220575450042753335235363, 3.98341487807363766257052150997, 4.88528464090276898990992310130, 6.51369476572287522616740385849, 7.915890555008525570426498062031, 8.163254117507368636335721399281, 8.595161165341223643221703117956, 9.836841586065077311185308875968, 10.99522020150026148888304460248

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