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Properties

Label 2-1815-5.4-c1-0-86
Degree $2$
Conductor $1815$
Sign $0.447 + 0.894i$
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  + 0.456i·2-s − i·3-s + 1.79·4-s + (2 − i)5-s + 0.456·6-s − 0.913i·7-s + 1.73i·8-s − 9-s + (0.456 + 0.913i)10-s − 1.79i·12-s − 0.913i·13-s + 0.417·14-s + (−1 − 2i)15-s + 2.79·16-s − 7.02i·17-s − 0.456i·18-s + ⋯
L(s)  = 1  + 0.323i·2-s − 0.577i·3-s + 0.895·4-s + (0.894 − 0.447i)5-s + 0.186·6-s − 0.345i·7-s + 0.612i·8-s − 0.333·9-s + (0.144 + 0.288i)10-s − 0.517i·12-s − 0.253i·13-s + 0.111·14-s + (−0.258 − 0.516i)15-s + 0.697·16-s − 1.70i·17-s − 0.107i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1815\)    =    \(3 \cdot 5 \cdot 11^{2}\)
Sign: $0.447 + 0.894i$
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: \((2,\ 1815,\ (\ :1/2),\ 0.447 + 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.456526466\)
\(L(\frac12)\) \(\approx\) \(2.456526466\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2 + i)T \)
11 \( 1 \)
good2 \( 1 - 0.456iT - 2T^{2} \)
7 \( 1 + 0.913iT - 7T^{2} \)
13 \( 1 + 0.913iT - 13T^{2} \)
17 \( 1 + 7.02iT - 17T^{2} \)
19 \( 1 + 5.29T + 19T^{2} \)
23 \( 1 + 0.582iT - 23T^{2} \)
29 \( 1 + 4.37T + 29T^{2} \)
31 \( 1 - 2.58T + 31T^{2} \)
37 \( 1 + 9.58iT - 37T^{2} \)
41 \( 1 - 6.92T + 41T^{2} \)
43 \( 1 - 4.37iT - 43T^{2} \)
47 \( 1 + 6.58iT - 47T^{2} \)
53 \( 1 - 5iT - 53T^{2} \)
59 \( 1 - 3.58T + 59T^{2} \)
61 \( 1 - 8.66T + 61T^{2} \)
67 \( 1 - 14.7iT - 67T^{2} \)
71 \( 1 + 9.16T + 71T^{2} \)
73 \( 1 - 15.6iT - 73T^{2} \)
79 \( 1 - 2.64T + 79T^{2} \)
83 \( 1 + 12.2iT - 83T^{2} \)
89 \( 1 + 16.7T + 89T^{2} \)
97 \( 1 - 3.58iT - 97T^{2} \)
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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

−9.012064706984668103239431095489, −8.269456073710762921090524588208, −7.26831728736222446694607900530, −6.91960571570504460687766867554, −5.88827769206638900969167950568, −5.44983194920763721162220063598, −4.25792188718505102566124520435, −2.73349674364345551979426968512, −2.14376968140260901998851661103, −0.872169443181384994440299486955, 1.64922955421527128658314865233, 2.38514807547878863481858209834, 3.37272544253470318171732258610, 4.30383098442226818716585037965, 5.58682326091720738960328554392, 6.20062183077121757660224346435, 6.75487103690445357925743615096, 7.912348039883231357643704726854, 8.744602707494548909963522821475, 9.613916301526506075783733368558

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