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Properties

Label 2-2151-717.716-c1-0-71
Degree $2$
Conductor $2151$
Sign $-0.832 + 0.553i$
Analytic cond. $17.1758$
Root an. cond. $4.14437$
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  − 0.525i·2-s + 1.72·4-s − 3.41i·5-s − 2.97i·7-s − 1.95i·8-s − 1.79·10-s − 3.57i·11-s + 3.39i·13-s − 1.56·14-s + 2.41·16-s − 2.65i·17-s + 0.176i·19-s − 5.87i·20-s − 1.88·22-s + 9.49·23-s + ⋯
L(s)  = 1  − 0.371i·2-s + 0.861·4-s − 1.52i·5-s − 1.12i·7-s − 0.691i·8-s − 0.566·10-s − 1.07i·11-s + 0.941i·13-s − 0.418·14-s + 0.604·16-s − 0.644i·17-s + 0.0403i·19-s − 1.31i·20-s − 0.401·22-s + 1.98·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 + 0.553i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.832 + 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2151\)    =    \(3^{2} \cdot 239\)
Sign: $-0.832 + 0.553i$
Analytic conductor: \(17.1758\)
Root analytic conductor: \(4.14437\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2151} (2150, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2151,\ (\ :1/2),\ -0.832 + 0.553i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.274107824\)
\(L(\frac12)\) \(\approx\) \(2.274107824\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
239 \( 1 + (-0.443 + 15.4i)T \)
good2 \( 1 + 0.525iT - 2T^{2} \)
5 \( 1 + 3.41iT - 5T^{2} \)
7 \( 1 + 2.97iT - 7T^{2} \)
11 \( 1 + 3.57iT - 11T^{2} \)
13 \( 1 - 3.39iT - 13T^{2} \)
17 \( 1 + 2.65iT - 17T^{2} \)
19 \( 1 - 0.176iT - 19T^{2} \)
23 \( 1 - 9.49T + 23T^{2} \)
29 \( 1 - 4.78iT - 29T^{2} \)
31 \( 1 - 1.35T + 31T^{2} \)
37 \( 1 + 0.737iT - 37T^{2} \)
41 \( 1 + 1.26T + 41T^{2} \)
43 \( 1 - 0.665iT - 43T^{2} \)
47 \( 1 - 7.68T + 47T^{2} \)
53 \( 1 + 5.98T + 53T^{2} \)
59 \( 1 + 11.2T + 59T^{2} \)
61 \( 1 - 3.71T + 61T^{2} \)
67 \( 1 + 13.2T + 67T^{2} \)
71 \( 1 + 1.99iT - 71T^{2} \)
73 \( 1 - 8.52iT - 73T^{2} \)
79 \( 1 - 6.23iT - 79T^{2} \)
83 \( 1 - 10.4iT - 83T^{2} \)
89 \( 1 - 6.78T + 89T^{2} \)
97 \( 1 - 15.1iT - 97T^{2} \)
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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

−8.990130313620843092537868565884, −7.988688508186757958885554691896, −7.18532674697829125322906267787, −6.54816909010900544480393089415, −5.46034940655263665441960531520, −4.67578356177263734329856869126, −3.80600668604120262749707076675, −2.85618301468112846021744703047, −1.38897893269388783707330168138, −0.829019534763027810759159180015, 1.85888009860240654908306399150, 2.74300918548361121686330832939, 3.19514798600595736824684561086, 4.74656649471712925196427283971, 5.79327410158197970057599467674, 6.24964238158629349512719448354, 7.14669845633419147697880803601, 7.52620838711189965484122194244, 8.464450768595745683007089035033, 9.413919275632566718084739594428

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