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Properties

Label 2-1815-5.4-c1-0-88
Degree $2$
Conductor $1815$
Sign $-0.405 - 0.914i$
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  − 2.42i·2-s − i·3-s − 3.90·4-s + (2.04 − 0.906i)5-s − 2.42·6-s + 1.34i·7-s + 4.61i·8-s − 9-s + (−2.20 − 4.96i)10-s + 3.90i·12-s − 4.12i·13-s + 3.25·14-s + (−0.906 − 2.04i)15-s + 3.41·16-s − 5.87i·17-s + 2.42i·18-s + ⋯
L(s)  = 1  − 1.71i·2-s − 0.577i·3-s − 1.95·4-s + (0.914 − 0.405i)5-s − 0.991·6-s + 0.507i·7-s + 1.63i·8-s − 0.333·9-s + (−0.696 − 1.57i)10-s + 1.12i·12-s − 1.14i·13-s + 0.871·14-s + (−0.234 − 0.527i)15-s + 0.853·16-s − 1.42i·17-s + 0.572i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.405 - 0.914i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.405 - 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1815\)    =    \(3 \cdot 5 \cdot 11^{2}\)
Sign: $-0.405 - 0.914i$
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: \((2,\ 1815,\ (\ :1/2),\ -0.405 - 0.914i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.517768946\)
\(L(\frac12)\) \(\approx\) \(1.517768946\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.04 + 0.906i)T \)
11 \( 1 \)
good2 \( 1 + 2.42iT - 2T^{2} \)
7 \( 1 - 1.34iT - 7T^{2} \)
13 \( 1 + 4.12iT - 13T^{2} \)
17 \( 1 + 5.87iT - 17T^{2} \)
19 \( 1 - 3.51T + 19T^{2} \)
23 \( 1 + 6.37iT - 23T^{2} \)
29 \( 1 + 6.39T + 29T^{2} \)
31 \( 1 - 9.40T + 31T^{2} \)
37 \( 1 - 7.01iT - 37T^{2} \)
41 \( 1 + 8.63T + 41T^{2} \)
43 \( 1 + 5.31iT - 43T^{2} \)
47 \( 1 - 3.74iT - 47T^{2} \)
53 \( 1 - 6.41iT - 53T^{2} \)
59 \( 1 + 8.97T + 59T^{2} \)
61 \( 1 + 6.52T + 61T^{2} \)
67 \( 1 - 8.83iT - 67T^{2} \)
71 \( 1 + 7.17T + 71T^{2} \)
73 \( 1 - 4.08iT - 73T^{2} \)
79 \( 1 + 9.97T + 79T^{2} \)
83 \( 1 + 3.06iT - 83T^{2} \)
89 \( 1 - 11.8T + 89T^{2} \)
97 \( 1 + 2.55iT - 97T^{2} \)
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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

−8.919077027972645959616306711217, −8.384789287543529995093963479940, −7.23053729944482641619170102097, −6.07909432583774855971870886646, −5.24211426717859684196868978859, −4.55829262372202484598846228033, −2.95961695212447679635591267739, −2.70825692144223964903939869562, −1.52884939862730352283365646184, −0.57852210938940195702362640073, 1.73785604442576138673921398258, 3.44805811069185965712021679613, 4.31384160204065105579696220611, 5.22410783853781190414213511464, 5.93408634387005805737074945869, 6.54947231785143411779306590905, 7.31417523825312234400374804693, 8.057626586924304885697383602785, 9.044160690557438517958838073028, 9.482488257139336754245546696997

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