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Properties

Label 2-1815-165.119-c0-0-3
Degree $2$
Conductor $1815$
Sign $0.998 + 0.0475i$
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  + (0.309 − 0.951i)2-s + (−0.809 + 0.587i)3-s + (0.309 + 0.951i)5-s + (0.309 + 0.951i)6-s + (0.809 − 0.587i)8-s + (0.309 − 0.951i)9-s + 0.999·10-s + (−0.809 − 0.587i)15-s + (−0.309 − 0.951i)16-s + (0.309 + 0.951i)17-s + (−0.809 − 0.587i)18-s + (1.61 − 1.17i)19-s − 23-s + (−0.309 + 0.951i)24-s + (−0.809 + 0.587i)25-s + ⋯
L(s)  = 1  + (0.309 − 0.951i)2-s + (−0.809 + 0.587i)3-s + (0.309 + 0.951i)5-s + (0.309 + 0.951i)6-s + (0.809 − 0.587i)8-s + (0.309 − 0.951i)9-s + 0.999·10-s + (−0.809 − 0.587i)15-s + (−0.309 − 0.951i)16-s + (0.309 + 0.951i)17-s + (−0.809 − 0.587i)18-s + (1.61 − 1.17i)19-s − 23-s + (−0.309 + 0.951i)24-s + (−0.809 + 0.587i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0475i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0475i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1815\)    =    \(3 \cdot 5 \cdot 11^{2}\)
Sign: $0.998 + 0.0475i$
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: \((2,\ 1815,\ (\ :0),\ 0.998 + 0.0475i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.284867707\)
\(L(\frac12)\) \(\approx\) \(1.284867707\)
\(L(1)\) 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 \( 1 + (-0.309 + 0.951i)T + (-0.809 - 0.587i)T^{2} \)
7 \( 1 + (-0.309 - 0.951i)T^{2} \)
13 \( 1 + (0.809 + 0.587i)T^{2} \)
17 \( 1 + (-0.309 - 0.951i)T + (-0.809 + 0.587i)T^{2} \)
19 \( 1 + (-1.61 + 1.17i)T + (0.309 - 0.951i)T^{2} \)
23 \( 1 + T + T^{2} \)
29 \( 1 + (-0.309 - 0.951i)T^{2} \)
31 \( 1 + (0.309 - 0.951i)T + (-0.809 - 0.587i)T^{2} \)
37 \( 1 + (-0.309 - 0.951i)T^{2} \)
41 \( 1 + (-0.309 + 0.951i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (-0.809 + 0.587i)T + (0.309 - 0.951i)T^{2} \)
53 \( 1 + (0.309 - 0.951i)T + (-0.809 - 0.587i)T^{2} \)
59 \( 1 + (-0.309 - 0.951i)T^{2} \)
61 \( 1 + (-0.309 - 0.951i)T + (-0.809 + 0.587i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (0.809 - 0.587i)T^{2} \)
73 \( 1 + (-0.309 - 0.951i)T^{2} \)
79 \( 1 + (-0.309 + 0.951i)T + (-0.809 - 0.587i)T^{2} \)
83 \( 1 + (0.618 + 1.90i)T + (-0.809 + 0.587i)T^{2} \)
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

−9.830165367069759099516286285641, −9.069221475284947836775151941417, −7.62492062303074324880199336365, −7.01691990095416094448918998811, −6.13978561580949642291744939102, −5.33605469270689964151766650535, −4.28200930458332768012529317622, −3.47953408273756752177002440160, −2.73863447088216778792899886212, −1.41821585640059455175789431370, 1.15716922919027064034339156068, 2.19887835494738640257567977711, 4.00285084248071983394417983302, 5.10126300999550843676915266983, 5.48275132470047319143408381648, 6.10857673654234470254613678878, 7.02509386974591116853975895106, 7.77560610656769679607964953582, 8.206580168450608336105137056588, 9.554117930769481350617675404112

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