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Properties

Label 2-1815-5.4-c1-0-97
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  + 2.18i·2-s − i·3-s − 2.79·4-s + (2 − i)5-s + 2.18·6-s − 4.37i·7-s − 1.73i·8-s − 9-s + (2.18 + 4.37i)10-s + 2.79i·12-s − 4.37i·13-s + 9.58·14-s + (−1 − 2i)15-s − 1.79·16-s − 3.55i·17-s − 2.18i·18-s + ⋯
L(s)  = 1  + 1.54i·2-s − 0.577i·3-s − 1.39·4-s + (0.894 − 0.447i)5-s + 0.893·6-s − 1.65i·7-s − 0.612i·8-s − 0.333·9-s + (0.692 + 1.38i)10-s + 0.805i·12-s − 1.21i·13-s + 2.56·14-s + (−0.258 − 0.516i)15-s − 0.447·16-s − 0.863i·17-s − 0.515i·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

\(L(1)\) \(\approx\) \(1.189164343\)
\(L(\frac12)\) \(\approx\) \(1.189164343\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.18iT - 2T^{2} \)
7 \( 1 + 4.37iT - 7T^{2} \)
13 \( 1 + 4.37iT - 13T^{2} \)
17 \( 1 + 3.55iT - 17T^{2} \)
19 \( 1 + 5.29T + 19T^{2} \)
23 \( 1 - 8.58iT - 23T^{2} \)
29 \( 1 + 0.913T + 29T^{2} \)
31 \( 1 + 6.58T + 31T^{2} \)
37 \( 1 + 0.417iT - 37T^{2} \)
41 \( 1 + 6.92T + 41T^{2} \)
43 \( 1 - 0.913iT - 43T^{2} \)
47 \( 1 - 2.58iT - 47T^{2} \)
53 \( 1 - 5iT - 53T^{2} \)
59 \( 1 + 5.58T + 59T^{2} \)
61 \( 1 + 8.66T + 61T^{2} \)
67 \( 1 + 12.7iT - 67T^{2} \)
71 \( 1 - 9.16T + 71T^{2} \)
73 \( 1 + 5.10iT - 73T^{2} \)
79 \( 1 - 2.64T + 79T^{2} \)
83 \( 1 - 1.63iT - 83T^{2} \)
89 \( 1 - 10.7T + 89T^{2} \)
97 \( 1 + 5.58iT - 97T^{2} \)
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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

−8.933690296368970015738607200053, −7.893899891280124989406451707867, −7.55574417094516416452910964927, −6.80369164717836451215404253556, −6.06722142226701153063678902660, −5.32267746046364799032502480822, −4.59140083342498773021894369266, −3.35810045947812111563884862118, −1.74070479175142017632947858453, −0.40740572924664099062540399203, 1.98338407681262515928577261901, 2.18206725836976204408133016842, 3.23786805469068201103229608061, 4.24432182623221976325615265098, 5.10868456080130736300456001525, 6.12295589574099612816892299568, 6.69098319090565357448893020322, 8.581668423962115981432932261462, 8.848454605347608471035954334559, 9.544288496925272077136204937515

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