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Properties

Label 2-2151-2151.1672-c0-0-2
Degree $2$
Conductor $2151$
Sign $-0.882 - 0.469i$
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  + (−0.438 + 0.759i)2-s + (0.913 + 0.406i)3-s + (0.115 + 0.200i)4-s + (0.241 + 0.419i)5-s + (−0.709 + 0.515i)6-s − 1.07·8-s + (0.669 + 0.743i)9-s − 0.424·10-s + (−0.848 + 1.46i)11-s + (0.0241 + 0.230i)12-s + (0.0505 + 0.481i)15-s + (0.357 − 0.619i)16-s + 0.347·17-s + (−0.857 + 0.182i)18-s + (−0.0559 + 0.0969i)20-s + ⋯
L(s)  = 1  + (−0.438 + 0.759i)2-s + (0.913 + 0.406i)3-s + (0.115 + 0.200i)4-s + (0.241 + 0.419i)5-s + (−0.709 + 0.515i)6-s − 1.07·8-s + (0.669 + 0.743i)9-s − 0.424·10-s + (−0.848 + 1.46i)11-s + (0.0241 + 0.230i)12-s + (0.0505 + 0.481i)15-s + (0.357 − 0.619i)16-s + 0.347·17-s + (−0.857 + 0.182i)18-s + (−0.0559 + 0.0969i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.882 - 0.469i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.882 - 0.469i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2151\)    =    \(3^{2} \cdot 239\)
Sign: $-0.882 - 0.469i$
Analytic conductor: \(1.07348\)
Root analytic conductor: \(1.03609\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2151} (1672, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2151,\ (\ :0),\ -0.882 - 0.469i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.284237914\)
\(L(\frac12)\) \(\approx\) \(1.284237914\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.913 - 0.406i)T \)
239 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (0.438 - 0.759i)T + (-0.5 - 0.866i)T^{2} \)
5 \( 1 + (-0.241 - 0.419i)T + (-0.5 + 0.866i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.848 - 1.46i)T + (-0.5 - 0.866i)T^{2} \)
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 \( 1 + (-0.374 + 0.648i)T + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.559 + 0.968i)T + (-0.5 + 0.866i)T^{2} \)
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 \( 1 + (0.990 - 1.71i)T + (-0.5 - 0.866i)T^{2} \)
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

−9.723856075058595569105434328179, −8.588103902323905331490022875721, −8.057064785485780826337741219551, −7.33340812660338371478515383409, −6.86646201572340759607548062463, −5.77569698368231567183260054367, −4.78662881393927172665873735903, −3.83800763611928686466138027519, −2.74712644973956166447684924914, −2.16069365745238328637778608804, 0.935104696090236372487120958617, 1.89109470362651810847858307578, 3.01594258365470416879128278832, 3.40652123014989240903238713033, 4.98134647448851282191186795700, 5.81426832190266595459614115699, 6.62884223247389634559786462450, 7.64819999821275001422797940878, 8.457475120979652644188716131563, 8.950686381963659900142165240200

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