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Properties

Label 2-2511-279.194-c0-0-2
Degree $2$
Conductor $2511$
Sign $0.994 - 0.107i$
Analytic cond. $1.25315$
Root an. cond. $1.11944$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.913 − 0.406i)4-s + (−0.0646 + 0.614i)7-s + (1.58 + 0.336i)13-s + (0.669 − 0.743i)16-s + (−0.5 + 1.53i)19-s + (−0.5 − 0.866i)25-s + (0.190 + 0.587i)28-s + (−0.978 − 0.207i)31-s + 0.618·37-s + (−0.604 + 0.128i)43-s + (0.604 + 0.128i)49-s + (1.58 − 0.336i)52-s + (0.809 + 1.40i)61-s + (0.309 − 0.951i)64-s + (0.809 − 1.40i)67-s + ⋯
L(s)  = 1  + (0.913 − 0.406i)4-s + (−0.0646 + 0.614i)7-s + (1.58 + 0.336i)13-s + (0.669 − 0.743i)16-s + (−0.5 + 1.53i)19-s + (−0.5 − 0.866i)25-s + (0.190 + 0.587i)28-s + (−0.978 − 0.207i)31-s + 0.618·37-s + (−0.604 + 0.128i)43-s + (0.604 + 0.128i)49-s + (1.58 − 0.336i)52-s + (0.809 + 1.40i)61-s + (0.309 − 0.951i)64-s + (0.809 − 1.40i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2511 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.107i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2511 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.107i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2511\)    =    \(3^{4} \cdot 31\)
Sign: $0.994 - 0.107i$
Analytic conductor: \(1.25315\)
Root analytic conductor: \(1.11944\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2511} (2240, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2511,\ (\ :0),\ 0.994 - 0.107i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.576935925\)
\(L(\frac12)\) \(\approx\) \(1.576935925\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 + (0.978 + 0.207i)T \)
good2 \( 1 + (-0.913 + 0.406i)T^{2} \)
5 \( 1 + (0.5 + 0.866i)T^{2} \)
7 \( 1 + (0.0646 - 0.614i)T + (-0.978 - 0.207i)T^{2} \)
11 \( 1 + (0.978 + 0.207i)T^{2} \)
13 \( 1 + (-1.58 - 0.336i)T + (0.913 + 0.406i)T^{2} \)
17 \( 1 + (-0.309 + 0.951i)T^{2} \)
19 \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \)
23 \( 1 + (0.978 - 0.207i)T^{2} \)
29 \( 1 + (-0.913 + 0.406i)T^{2} \)
37 \( 1 - 0.618T + T^{2} \)
41 \( 1 + (0.104 - 0.994i)T^{2} \)
43 \( 1 + (0.604 - 0.128i)T + (0.913 - 0.406i)T^{2} \)
47 \( 1 + (0.104 + 0.994i)T^{2} \)
53 \( 1 + (-0.309 + 0.951i)T^{2} \)
59 \( 1 + (-0.913 - 0.406i)T^{2} \)
61 \( 1 + (-0.809 - 1.40i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.809 + 1.40i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.309 + 0.951i)T^{2} \)
73 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
79 \( 1 + (1.47 + 0.658i)T + (0.669 + 0.743i)T^{2} \)
83 \( 1 + (-0.913 + 0.406i)T^{2} \)
89 \( 1 + (-0.309 - 0.951i)T^{2} \)
97 \( 1 + (-0.169 + 1.60i)T + (-0.978 - 0.207i)T^{2} \)
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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

−9.041487777968282032889643332920, −8.340178124427107298898333060305, −7.62655349119369849603069747357, −6.60161955608256762584557416876, −6.00634274597592826800207246847, −5.58074605114162306643488166935, −4.20732303652218136339670605428, −3.38310302564736269719904768144, −2.24538934467737999414777748839, −1.44065626501660578239708976419, 1.23476974978811524007596845146, 2.41126854135560080618678913756, 3.44636998605109234698239128422, 4.02683623167074141971549605896, 5.28494165079099428490135367640, 6.15187920536908599334463016113, 6.86632237952604793177922227647, 7.42856396028134791351873563650, 8.325061023126132332821516663861, 8.905350474563508115549681578241

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