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Properties

Label 2-2151-2151.238-c0-0-12
Degree $2$
Conductor $2151$
Sign $0.990 + 0.139i$
Analytic cond. $1.07348$
Root an. cond. $1.03609$
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  + (0.719 + 1.24i)2-s + (−0.104 − 0.994i)3-s + (−0.534 + 0.926i)4-s + (0.997 − 1.72i)5-s + (1.16 − 0.845i)6-s − 0.100·8-s + (−0.978 + 0.207i)9-s + 2.87·10-s + (0.374 + 0.648i)11-s + (0.977 + 0.435i)12-s + (−1.82 − 0.811i)15-s + (0.462 + 0.801i)16-s + 0.347·17-s + (−0.962 − 1.06i)18-s + (1.06 + 1.84i)20-s + ⋯
L(s)  = 1  + (0.719 + 1.24i)2-s + (−0.104 − 0.994i)3-s + (−0.534 + 0.926i)4-s + (0.997 − 1.72i)5-s + (1.16 − 0.845i)6-s − 0.100·8-s + (−0.978 + 0.207i)9-s + 2.87·10-s + (0.374 + 0.648i)11-s + (0.977 + 0.435i)12-s + (−1.82 − 0.811i)15-s + (0.462 + 0.801i)16-s + 0.347·17-s + (−0.962 − 1.06i)18-s + (1.06 + 1.84i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.139i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.139i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2151\)    =    \(3^{2} \cdot 239\)
Sign: $0.990 + 0.139i$
Analytic conductor: \(1.07348\)
Root analytic conductor: \(1.03609\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2151} (238, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2151,\ (\ :0),\ 0.990 + 0.139i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.973955769\)
\(L(\frac12)\) \(\approx\) \(1.973955769\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.104 + 0.994i)T \)
239 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (-0.719 - 1.24i)T + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + (-0.997 + 1.72i)T + (-0.5 - 0.866i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.374 - 0.648i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 - 0.347T + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.848 + 1.46i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.961 - 1.66i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 - 0.618T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.882 - 1.52i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.5 - 0.866i)T^{2} \)
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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

−9.039648963825575934556290616090, −8.154736968134464318346814747265, −7.68713755864819826759685226929, −6.65515343615949225531721648931, −6.12690071607839028743575783672, −5.30376313681317690609111543065, −4.96334794275635969081449843893, −3.89076934471466921933478666850, −2.09419898732789647147549954291, −1.28513377143557202752403216467, 1.85169019732102137321684816579, 2.76454031636809279004378509639, 3.42805355450789153828449672968, 3.96809230738928542715019338230, 5.32280337248662310963812374188, 5.76238246199488826113256529223, 6.70899504128218267763661069985, 7.65025163872624006985384437580, 9.018072252947797977269237137583, 9.716116344506322578271487281829

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