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Properties

Label 2-1815-5.4-c1-0-43
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

\(L(1)\) \(\approx\) \(1.197467529\)
\(L(\frac12)\) \(\approx\) \(1.197467529\)
\(L(\frac{3}{2})\) 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

−9.142519585815445671917462386984, −8.593233828873155088676807514752, −7.82752314568611490000484241821, −6.59889357968233645851573589727, −5.34951843867878760892404626518, −4.63735225448071066959887585830, −3.69169634876992883646734444719, −3.19850370679296967751286757128, −2.04053441011335020925441395997, −0.66108526969250504655767240954, 0.845017002341672164817891231857, 2.77508826199097127036505583954, 3.91395112684317247676094501088, 4.82697747287156240031603121536, 5.65611290364373653423960513150, 6.60430384105414102218048295134, 7.36654492037850015869882076903, 7.48594106873973863377701040219, 8.313941119485027286048763348307, 9.176300092098749994898912685058

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