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Properties

Label 2-1815-5.4-c1-0-99
Degree $2$
Conductor $1815$
Sign $0.100 - 0.994i$
Analytic cond. $14.4928$
Root an. cond. $3.80694$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.41i·2-s − i·3-s + (−2.22 − 0.224i)5-s − 1.41·6-s + 1.73i·7-s − 2.82i·8-s − 9-s + (−0.317 + 3.14i)10-s + 0.635i·13-s + 2.44·14-s + (−0.224 + 2.22i)15-s − 4.00·16-s − 4.87i·17-s + 1.41i·18-s − 1.73·19-s + ⋯
L(s)  = 1  − 0.999i·2-s − 0.577i·3-s + (−0.994 − 0.100i)5-s − 0.577·6-s + 0.654i·7-s − 0.999i·8-s − 0.333·9-s + (−0.100 + 0.994i)10-s + 0.176i·13-s + 0.654·14-s + (−0.0580 + 0.574i)15-s − 1.00·16-s − 1.18i·17-s + 0.333i·18-s − 0.397·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.100 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.100 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1815\)    =    \(3 \cdot 5 \cdot 11^{2}\)
Sign: $0.100 - 0.994i$
Analytic conductor: \(14.4928\)
Root analytic conductor: \(3.80694\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1815} (364, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1815,\ (\ :1/2),\ 0.100 - 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1946319840\)
\(L(\frac12)\) \(\approx\) \(0.1946319840\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + iT \)
5 \( 1 + (2.22 + 0.224i)T \)
11 \( 1 \)
good2 \( 1 + 1.41iT - 2T^{2} \)
7 \( 1 - 1.73iT - 7T^{2} \)
13 \( 1 - 0.635iT - 13T^{2} \)
17 \( 1 + 4.87iT - 17T^{2} \)
19 \( 1 + 1.73T + 19T^{2} \)
23 \( 1 + 2iT - 23T^{2} \)
29 \( 1 + 8.34T + 29T^{2} \)
31 \( 1 + 7.89T + 31T^{2} \)
37 \( 1 - 5.89iT - 37T^{2} \)
41 \( 1 + 9.12T + 41T^{2} \)
43 \( 1 - 5.65iT - 43T^{2} \)
47 \( 1 - 7.34iT - 47T^{2} \)
53 \( 1 - 10iT - 53T^{2} \)
59 \( 1 - 12.2T + 59T^{2} \)
61 \( 1 + 3.28T + 61T^{2} \)
67 \( 1 + 4.10iT - 67T^{2} \)
71 \( 1 + 3.55T + 71T^{2} \)
73 \( 1 + 3.28iT - 73T^{2} \)
79 \( 1 + 6.75T + 79T^{2} \)
83 \( 1 + 12.7iT - 83T^{2} \)
89 \( 1 + 1.34T + 89T^{2} \)
97 \( 1 + 5iT - 97T^{2} \)
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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

−8.869700044463467707586853721638, −7.78884531090869001483752850191, −7.19473650836116509728066354065, −6.41008921563623100732948561975, −5.29810773043594528040369993060, −4.25529543890500113715571276823, −3.29341910904244448543649154248, −2.52975996585987316940892054618, −1.45479158454637593924667117321, −0.06884706550698680983907528386, 2.00964183851375818118520656580, 3.58716396980870253639803803814, 3.99091672962575069794673099996, 5.20569538519282428365327642960, 5.81169378898994196653469006961, 7.05312864249542716014402791312, 7.24191457720434818291741435078, 8.322788494012561404209994670611, 8.642788225643903483140752125310, 9.837980325892460426121310935723

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