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Properties

Label 2-1815-5.4-c1-0-47
Degree $2$
Conductor $1815$
Sign $0.288 - 0.957i$
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  + 2.21i·2-s + i·3-s − 2.90·4-s + (−2.14 − 0.644i)5-s − 2.21·6-s − 1.59i·7-s − 1.99i·8-s − 9-s + (1.42 − 4.74i)10-s − 2.90i·12-s + 0.891i·13-s + 3.52·14-s + (0.644 − 2.14i)15-s − 1.38·16-s − 4.43i·17-s − 2.21i·18-s + ⋯
L(s)  = 1  + 1.56i·2-s + 0.577i·3-s − 1.45·4-s + (−0.957 − 0.288i)5-s − 0.903·6-s − 0.602i·7-s − 0.705i·8-s − 0.333·9-s + (0.451 − 1.49i)10-s − 0.837i·12-s + 0.247i·13-s + 0.942·14-s + (0.166 − 0.552i)15-s − 0.346·16-s − 1.07i·17-s − 0.521i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.288 - 0.957i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.288 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1815\)    =    \(3 \cdot 5 \cdot 11^{2}\)
Sign: $0.288 - 0.957i$
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: \((2,\ 1815,\ (\ :1/2),\ 0.288 - 0.957i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.003746466\)
\(L(\frac12)\) \(\approx\) \(1.003746466\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.14 + 0.644i)T \)
11 \( 1 \)
good2 \( 1 - 2.21iT - 2T^{2} \)
7 \( 1 + 1.59iT - 7T^{2} \)
13 \( 1 - 0.891iT - 13T^{2} \)
17 \( 1 + 4.43iT - 17T^{2} \)
19 \( 1 + 5.84T + 19T^{2} \)
23 \( 1 + 3.27iT - 23T^{2} \)
29 \( 1 - 5.30T + 29T^{2} \)
31 \( 1 - 8.91T + 31T^{2} \)
37 \( 1 - 5.53iT - 37T^{2} \)
41 \( 1 - 4.19T + 41T^{2} \)
43 \( 1 - 6.39iT - 43T^{2} \)
47 \( 1 + 8.47iT - 47T^{2} \)
53 \( 1 + 4.38iT - 53T^{2} \)
59 \( 1 - 7.20T + 59T^{2} \)
61 \( 1 + 2.03T + 61T^{2} \)
67 \( 1 + 13.6iT - 67T^{2} \)
71 \( 1 - 1.15T + 71T^{2} \)
73 \( 1 - 6.66iT - 73T^{2} \)
79 \( 1 + 4.89T + 79T^{2} \)
83 \( 1 - 13.2iT - 83T^{2} \)
89 \( 1 - 8.06T + 89T^{2} \)
97 \( 1 - 10.7iT - 97T^{2} \)
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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

−9.092161884966425341822690194945, −8.285891712137680245082918281245, −8.018830365539484460979849705747, −6.82642643207491748235288914646, −6.60620060147157954597845758653, −5.32555684121660256572717415813, −4.51327020399175223804815315682, −4.19492390449208131469033748041, −2.78668318835782615007035051770, −0.50261670631009636377871299615, 0.910348025927798325915538800984, 2.17938111498354657510995766348, 2.88486422975356359618391727314, 3.87881637199076401425757862784, 4.55226822555598903373295377465, 5.89614529300377531201318103238, 6.72420883860359952561033967279, 7.77214399531903147153074506287, 8.523351282062490041041105768839, 9.061480251209433669847926459698

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