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Properties

Label 2-1815-165.119-c0-0-5
Degree $2$
Conductor $1815$
Sign $0.957 - 0.288i$
Analytic cond. $0.905802$
Root an. cond. $0.951736$
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  + (0.809 − 0.587i)3-s + (0.809 + 0.587i)4-s + (0.309 + 0.951i)5-s + (0.309 − 0.951i)9-s + 12-s + (0.809 + 0.587i)15-s + (0.309 + 0.951i)16-s + (−0.309 + 0.951i)20-s − 2·23-s + (−0.809 + 0.587i)25-s + (−0.309 − 0.951i)27-s + (0.618 − 1.90i)31-s + (0.809 − 0.587i)36-s + 45-s + (−1.61 + 1.17i)47-s + (0.809 + 0.587i)48-s + ⋯
L(s)  = 1  + (0.809 − 0.587i)3-s + (0.809 + 0.587i)4-s + (0.309 + 0.951i)5-s + (0.309 − 0.951i)9-s + 12-s + (0.809 + 0.587i)15-s + (0.309 + 0.951i)16-s + (−0.309 + 0.951i)20-s − 2·23-s + (−0.809 + 0.587i)25-s + (−0.309 − 0.951i)27-s + (0.618 − 1.90i)31-s + (0.809 − 0.587i)36-s + 45-s + (−1.61 + 1.17i)47-s + (0.809 + 0.587i)48-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.288i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.288i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1815\)    =    \(3 \cdot 5 \cdot 11^{2}\)
Sign: $0.957 - 0.288i$
Analytic conductor: \(0.905802\)
Root analytic conductor: \(0.951736\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1815} (614, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1815,\ (\ :0),\ 0.957 - 0.288i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.840662764\)
\(L(\frac12)\) \(\approx\) \(1.840662764\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.809 + 0.587i)T \)
5 \( 1 + (-0.309 - 0.951i)T \)
11 \( 1 \)
good2 \( 1 + (-0.809 - 0.587i)T^{2} \)
7 \( 1 + (-0.309 - 0.951i)T^{2} \)
13 \( 1 + (0.809 + 0.587i)T^{2} \)
17 \( 1 + (-0.809 + 0.587i)T^{2} \)
19 \( 1 + (0.309 - 0.951i)T^{2} \)
23 \( 1 + 2T + T^{2} \)
29 \( 1 + (-0.309 - 0.951i)T^{2} \)
31 \( 1 + (-0.618 + 1.90i)T + (-0.809 - 0.587i)T^{2} \)
37 \( 1 + (-0.309 - 0.951i)T^{2} \)
41 \( 1 + (-0.309 + 0.951i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (1.61 - 1.17i)T + (0.309 - 0.951i)T^{2} \)
53 \( 1 + (-0.618 + 1.90i)T + (-0.809 - 0.587i)T^{2} \)
59 \( 1 + (-0.309 - 0.951i)T^{2} \)
61 \( 1 + (-0.809 + 0.587i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (0.809 - 0.587i)T^{2} \)
73 \( 1 + (-0.309 - 0.951i)T^{2} \)
79 \( 1 + (-0.809 - 0.587i)T^{2} \)
83 \( 1 + (-0.809 + 0.587i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.809 + 0.587i)T^{2} \)
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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

−9.639425621504580191521693573095, −8.378314304960135305199617144675, −7.85804492829460603653058399781, −7.25104127957423472371838856686, −6.35556549788546608468668614745, −5.99605176218627479266010625220, −4.15684443637186762001407951090, −3.38144855573687194262805480722, −2.49155590994682791374078139547, −1.86586363846880906882406590842, 1.51730003409549472460613265647, 2.34137705233346790571998974515, 3.49856094264172595686549764941, 4.54857317189007985036241576170, 5.30329429578696702667737345503, 6.11449448705083067891280772583, 7.10900269519336955703568354151, 8.073574926583927336260868079150, 8.619669449526333634203592261444, 9.511232506295505346981006049401

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