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Properties

Label 2-1815-165.107-c0-0-1
Degree $2$
Conductor $1815$
Sign $-0.0983 + 0.995i$
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  + (−0.951 + 0.309i)3-s + (−0.951 − 0.309i)4-s + (0.587 − 0.809i)5-s + (0.809 − 0.587i)9-s + 0.999·12-s + (−0.309 + 0.951i)15-s + (0.809 + 0.587i)16-s + (−0.809 + 0.587i)20-s + (−1 − i)23-s + (−0.309 − 0.951i)25-s + (−0.587 + 0.809i)27-s + (−0.951 + 0.309i)36-s + (1.26 − 0.642i)37-s − 0.999i·45-s + (−1.26 − 0.642i)47-s + (−0.951 − 0.309i)48-s + ⋯
L(s)  = 1  + (−0.951 + 0.309i)3-s + (−0.951 − 0.309i)4-s + (0.587 − 0.809i)5-s + (0.809 − 0.587i)9-s + 0.999·12-s + (−0.309 + 0.951i)15-s + (0.809 + 0.587i)16-s + (−0.809 + 0.587i)20-s + (−1 − i)23-s + (−0.309 − 0.951i)25-s + (−0.587 + 0.809i)27-s + (−0.951 + 0.309i)36-s + (1.26 − 0.642i)37-s − 0.999i·45-s + (−1.26 − 0.642i)47-s + (−0.951 − 0.309i)48-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0983 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0983 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1815\)    =    \(3 \cdot 5 \cdot 11^{2}\)
Sign: $-0.0983 + 0.995i$
Analytic conductor: \(0.905802\)
Root analytic conductor: \(0.951736\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1815} (602, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1815,\ (\ :0),\ -0.0983 + 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5957511365\)
\(L(\frac12)\) \(\approx\) \(0.5957511365\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.951 - 0.309i)T \)
5 \( 1 + (-0.587 + 0.809i)T \)
11 \( 1 \)
good2 \( 1 + (0.951 + 0.309i)T^{2} \)
7 \( 1 + (-0.587 + 0.809i)T^{2} \)
13 \( 1 + (0.951 + 0.309i)T^{2} \)
17 \( 1 + (-0.951 + 0.309i)T^{2} \)
19 \( 1 + (-0.809 + 0.587i)T^{2} \)
23 \( 1 + (1 + i)T + iT^{2} \)
29 \( 1 + (0.809 + 0.587i)T^{2} \)
31 \( 1 + (0.309 - 0.951i)T^{2} \)
37 \( 1 + (-1.26 + 0.642i)T + (0.587 - 0.809i)T^{2} \)
41 \( 1 + (-0.809 + 0.587i)T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + (1.26 + 0.642i)T + (0.587 + 0.809i)T^{2} \)
53 \( 1 + (1.39 + 0.221i)T + (0.951 + 0.309i)T^{2} \)
59 \( 1 + (-0.618 + 1.90i)T + (-0.809 - 0.587i)T^{2} \)
61 \( 1 + (-0.309 - 0.951i)T^{2} \)
67 \( 1 + (1 + i)T + iT^{2} \)
71 \( 1 + (-0.309 - 0.951i)T^{2} \)
73 \( 1 + (0.587 - 0.809i)T^{2} \)
79 \( 1 + (0.309 - 0.951i)T^{2} \)
83 \( 1 + (0.951 - 0.309i)T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (-1.39 - 0.221i)T + (0.951 + 0.309i)T^{2} \)
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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

−9.447944267177284657813850202583, −8.624115256199519337933540337260, −7.85272693270942384891575024201, −6.46650145997521812112470110078, −5.94648421403701664063215157280, −5.06764319560394195040439528435, −4.59066910245693895161867185676, −3.70634128792935696105403447811, −1.87094488542955974047843283110, −0.55571399239884087171757280301, 1.44460981245747731966906226481, 2.83744244084766448925738654197, 3.98413574984624270876734025212, 4.84717528478467725278574719087, 5.78759624923262161428014286104, 6.26237064571582565579422503139, 7.38244547487673868966320706736, 7.85450237394018362141604887919, 9.015676138909973883103802082579, 9.864407645272558051232766844844

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