Properties

Label 2-1815-165.17-c0-0-0
Degree $2$
Conductor $1815$
Sign $0.120 - 0.992i$
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 + (0.587 + 0.809i)4-s + (−0.951 − 0.309i)5-s + (0.309 + 0.951i)9-s + i·12-s + (−0.587 − 0.809i)15-s + (−0.309 + 0.951i)16-s + (−0.309 − 0.951i)20-s + (1 + i)23-s + (0.809 + 0.587i)25-s + (−0.309 + 0.951i)27-s + (−0.587 + 0.809i)36-s + (−1.39 − 0.221i)37-s i·45-s + (−1.39 + 0.221i)47-s + (−0.809 + 0.587i)48-s + ⋯
L(s)  = 1  + (0.809 + 0.587i)3-s + (0.587 + 0.809i)4-s + (−0.951 − 0.309i)5-s + (0.309 + 0.951i)9-s + i·12-s + (−0.587 − 0.809i)15-s + (−0.309 + 0.951i)16-s + (−0.309 − 0.951i)20-s + (1 + i)23-s + (0.809 + 0.587i)25-s + (−0.309 + 0.951i)27-s + (−0.587 + 0.809i)36-s + (−1.39 − 0.221i)37-s i·45-s + (−1.39 + 0.221i)47-s + (−0.809 + 0.587i)48-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1815\)    =    \(3 \cdot 5 \cdot 11^{2}\)
Sign: $0.120 - 0.992i$
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: \((2,\ 1815,\ (\ :0),\ 0.120 - 0.992i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.436957853\)
\(L(\frac12)\) \(\approx\) \(1.436957853\)
\(L(1)\) 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 \( 1 + (0.951 + 0.309i)T \)
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

−9.442907020808774020835200208925, −8.655567003840483833082946726629, −8.190292800040344033451663153453, −7.37773190266209550570011825782, −6.88057781506889136015895894942, −5.37715892713181366543485477392, −4.49254381980272329292924061391, −3.57374496719152881985784435603, −3.13828919117285998981495708853, −1.86251019477020712152648342432, 1.03488879542574869347554384729, 2.34095629303386294189897794133, 3.12859673559292412674285287669, 4.16719298543467603491519853404, 5.24843999553029490792212808672, 6.40881917787331710919119513636, 6.97882034810401776541886556228, 7.54441120654552481840052216170, 8.520876058625403629877583049485, 9.054707042082629008026933393506

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