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Properties

Label 2-460-115.102-c1-0-4
Degree $2$
Conductor $460$
Sign $0.803 - 0.594i$
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  + (−1.88 + 0.409i)3-s + (1.87 + 1.21i)5-s + (2.30 − 1.26i)7-s + (0.641 − 0.292i)9-s + (−1.17 − 1.01i)11-s + (0.559 − 1.02i)13-s + (−4.02 − 1.52i)15-s + (4.73 + 3.54i)17-s + (−0.0509 + 0.354i)19-s + (−3.82 + 3.31i)21-s + (1.64 + 4.50i)23-s + (2.03 + 4.56i)25-s + (3.53 − 2.64i)27-s + (3.90 − 0.561i)29-s + (−2.13 + 1.37i)31-s + ⋯
L(s)  = 1  + (−1.08 + 0.236i)3-s + (0.838 + 0.544i)5-s + (0.873 − 0.476i)7-s + (0.213 − 0.0976i)9-s + (−0.354 − 0.307i)11-s + (0.155 − 0.284i)13-s + (−1.03 − 0.393i)15-s + (1.14 + 0.860i)17-s + (−0.0116 + 0.0812i)19-s + (−0.835 + 0.723i)21-s + (0.342 + 0.939i)23-s + (0.406 + 0.913i)25-s + (0.680 − 0.509i)27-s + (0.725 − 0.104i)29-s + (−0.383 + 0.246i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.803 - 0.594i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.803 - 0.594i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $0.803 - 0.594i$
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: \((2,\ 460,\ (\ :1/2),\ 0.803 - 0.594i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.15881 + 0.382043i\)
\(L(\frac12)\) \(\approx\) \(1.15881 + 0.382043i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.87 - 1.21i)T \)
23 \( 1 + (-1.64 - 4.50i)T \)
good3 \( 1 + (1.88 - 0.409i)T + (2.72 - 1.24i)T^{2} \)
7 \( 1 + (-2.30 + 1.26i)T + (3.78 - 5.88i)T^{2} \)
11 \( 1 + (1.17 + 1.01i)T + (1.56 + 10.8i)T^{2} \)
13 \( 1 + (-0.559 + 1.02i)T + (-7.02 - 10.9i)T^{2} \)
17 \( 1 + (-4.73 - 3.54i)T + (4.78 + 16.3i)T^{2} \)
19 \( 1 + (0.0509 - 0.354i)T + (-18.2 - 5.35i)T^{2} \)
29 \( 1 + (-3.90 + 0.561i)T + (27.8 - 8.17i)T^{2} \)
31 \( 1 + (2.13 - 1.37i)T + (12.8 - 28.1i)T^{2} \)
37 \( 1 + (-8.38 - 3.12i)T + (27.9 + 24.2i)T^{2} \)
41 \( 1 + (2.24 - 4.92i)T + (-26.8 - 30.9i)T^{2} \)
43 \( 1 + (1.22 + 5.64i)T + (-39.1 + 17.8i)T^{2} \)
47 \( 1 + (-5.03 - 5.03i)T + 47iT^{2} \)
53 \( 1 + (-1.91 - 3.51i)T + (-28.6 + 44.5i)T^{2} \)
59 \( 1 + (3.29 + 11.2i)T + (-49.6 + 31.8i)T^{2} \)
61 \( 1 + (1.61 + 2.51i)T + (-25.3 + 55.4i)T^{2} \)
67 \( 1 + (11.7 + 0.838i)T + (66.3 + 9.53i)T^{2} \)
71 \( 1 + (-1.27 - 1.47i)T + (-10.1 + 70.2i)T^{2} \)
73 \( 1 + (0.0365 + 0.0488i)T + (-20.5 + 70.0i)T^{2} \)
79 \( 1 + (4.71 - 1.38i)T + (66.4 - 42.7i)T^{2} \)
83 \( 1 + (-5.17 + 13.8i)T + (-62.7 - 54.3i)T^{2} \)
89 \( 1 + (12.6 + 8.15i)T + (36.9 + 80.9i)T^{2} \)
97 \( 1 + (3.09 + 8.30i)T + (-73.3 + 63.5i)T^{2} \)
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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

−10.97295866694014395918102973498, −10.50300785951957103872225916373, −9.708284325136983998974503278538, −8.324233710217585705857511293621, −7.42852532198118416492180124897, −6.15624719912241294651966550480, −5.60922921830544877822464784839, −4.65736117449561424181466554488, −3.11979584989111627707042476614, −1.36996804783018561534854460022, 1.05905580322518442115265928721, 2.52986263968774500257493576899, 4.62785652522057447024687869604, 5.34694361721989501076023002837, 6.01867079205683383907230879673, 7.13663502843896978555504984030, 8.313964242949157585689788240226, 9.196632656290049994167285574873, 10.20210778816838223070317926286, 11.05072772793139602636357969556

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