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Properties

Label 2-1815-5.4-c1-0-107
Degree $2$
Conductor $1815$
Sign $0.999 - 0.0426i$
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  − 2.59i·2-s − i·3-s − 4.71·4-s + (−0.0953 − 2.23i)5-s − 2.59·6-s − 3.41i·7-s + 7.04i·8-s − 9-s + (−5.79 + 0.247i)10-s + 4.71i·12-s + 4.58i·13-s − 8.86·14-s + (−2.23 + 0.0953i)15-s + 8.82·16-s − 1.97i·17-s + 2.59i·18-s + ⋯
L(s)  = 1  − 1.83i·2-s − 0.577i·3-s − 2.35·4-s + (−0.0426 − 0.999i)5-s − 1.05·6-s − 1.29i·7-s + 2.49i·8-s − 0.333·9-s + (−1.83 + 0.0781i)10-s + 1.36i·12-s + 1.27i·13-s − 2.36·14-s + (−0.576 + 0.0246i)15-s + 2.20·16-s − 0.479i·17-s + 0.610i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0426i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0426i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1815\)    =    \(3 \cdot 5 \cdot 11^{2}\)
Sign: $0.999 - 0.0426i$
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: \((2,\ 1815,\ (\ :1/2),\ 0.999 - 0.0426i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4159330423\)
\(L(\frac12)\) \(\approx\) \(0.4159330423\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.0953 + 2.23i)T \)
11 \( 1 \)
good2 \( 1 + 2.59iT - 2T^{2} \)
7 \( 1 + 3.41iT - 7T^{2} \)
13 \( 1 - 4.58iT - 13T^{2} \)
17 \( 1 + 1.97iT - 17T^{2} \)
19 \( 1 + 1.41T + 19T^{2} \)
23 \( 1 - 5.03iT - 23T^{2} \)
29 \( 1 + 4.34T + 29T^{2} \)
31 \( 1 + 0.273T + 31T^{2} \)
37 \( 1 - 0.245iT - 37T^{2} \)
41 \( 1 - 1.33T + 41T^{2} \)
43 \( 1 + 11.7iT - 43T^{2} \)
47 \( 1 - 12.9iT - 47T^{2} \)
53 \( 1 - 0.972iT - 53T^{2} \)
59 \( 1 + 6.09T + 59T^{2} \)
61 \( 1 + 7.80T + 61T^{2} \)
67 \( 1 - 2.14iT - 67T^{2} \)
71 \( 1 + 9.09T + 71T^{2} \)
73 \( 1 + 12.6iT - 73T^{2} \)
79 \( 1 + 0.769T + 79T^{2} \)
83 \( 1 + 9.73iT - 83T^{2} \)
89 \( 1 + 10.2T + 89T^{2} \)
97 \( 1 + 13.6iT - 97T^{2} \)
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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

−8.991725014610454667186955601042, −7.84351043114483711843449035200, −7.15485994079669143938974050786, −5.80328729826616393679121157701, −4.65616863618106850134527780612, −4.18761442192991867413869419816, −3.29255433555866251726641704356, −1.91717430355737382010055868497, −1.29560678819910228117399367563, −0.16398223677416275893277078870, 2.56932551301137615903159317227, 3.59762928136376503874410862346, 4.66243686394487334482840258640, 5.59044241336032980338452882849, 5.99148821866652714960034321165, 6.74336981594295037579938141504, 7.72413923148258763501753711546, 8.316114137913432813469591739459, 8.955725552196229551559105659669, 9.795239250493808204156489263421

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