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Properties

Label 2-1815-165.17-c0-0-1
Degree $2$
Conductor $1815$
Sign $0.992 + 0.120i$
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  + (0.587 − 0.809i)3-s + (0.587 + 0.809i)4-s + (0.951 + 0.309i)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 + (−1 − i)23-s + (0.809 + 0.587i)25-s + (−0.951 − 0.309i)27-s + (0.587 − 0.809i)36-s + (−1.39 − 0.221i)37-s − i·45-s + (1.39 − 0.221i)47-s + (0.587 + 0.809i)48-s + ⋯
L(s)  = 1  + (0.587 − 0.809i)3-s + (0.587 + 0.809i)4-s + (0.951 + 0.309i)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 + (−1 − i)23-s + (0.809 + 0.587i)25-s + (−0.951 − 0.309i)27-s + (0.587 − 0.809i)36-s + (−1.39 − 0.221i)37-s − i·45-s + (1.39 − 0.221i)47-s + (0.587 + 0.809i)48-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.120i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.120i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1815\)    =    \(3 \cdot 5 \cdot 11^{2}\)
Sign: $0.992 + 0.120i$
Analytic conductor: \(0.905802\)
Root analytic conductor: \(0.951736\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1815} (1667, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1815,\ (\ :0),\ 0.992 + 0.120i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.788253360\)
\(L(\frac12)\) \(\approx\) \(1.788253360\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.587 + 0.809i)T \)
5 \( 1 + (-0.951 - 0.309i)T \)
11 \( 1 \)
good2 \( 1 + (-0.587 - 0.809i)T^{2} \)
7 \( 1 + (-0.951 - 0.309i)T^{2} \)
13 \( 1 + (-0.587 - 0.809i)T^{2} \)
17 \( 1 + (0.587 - 0.809i)T^{2} \)
19 \( 1 + (0.309 + 0.951i)T^{2} \)
23 \( 1 + (1 + i)T + iT^{2} \)
29 \( 1 + (-0.309 + 0.951i)T^{2} \)
31 \( 1 + (-0.809 + 0.587i)T^{2} \)
37 \( 1 + (1.39 + 0.221i)T + (0.951 + 0.309i)T^{2} \)
41 \( 1 + (0.309 + 0.951i)T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + (-1.39 + 0.221i)T + (0.951 - 0.309i)T^{2} \)
53 \( 1 + (0.642 - 1.26i)T + (-0.587 - 0.809i)T^{2} \)
59 \( 1 + (1.61 - 1.17i)T + (0.309 - 0.951i)T^{2} \)
61 \( 1 + (0.809 + 0.587i)T^{2} \)
67 \( 1 + (1 + i)T + iT^{2} \)
71 \( 1 + (0.809 + 0.587i)T^{2} \)
73 \( 1 + (0.951 + 0.309i)T^{2} \)
79 \( 1 + (-0.809 + 0.587i)T^{2} \)
83 \( 1 + (-0.587 + 0.809i)T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (-0.642 + 1.26i)T + (-0.587 - 0.809i)T^{2} \)
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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

−9.132439968704015608647037316237, −8.692626391584907674051777744619, −7.71517174689961644107195917205, −7.17178969833146395114178261431, −6.35310951219596576847330566789, −5.80811663667059189441637376898, −4.29354903793552487199441829448, −3.20022035866513305054314161412, −2.47213697560401023017591924287, −1.66253093286661125159046525303, 1.62244923440800475987779718734, 2.40983579900300716172195148637, 3.52366881732014840383831392483, 4.71041259104091355513745251200, 5.44549072238680958288104697110, 6.04954123423968237103988564803, 7.06281275344339944165591053550, 8.014502542511690357741725639752, 8.990247853083397431982424842278, 9.501036892048083563027052913293

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