Properties

Label 2-1815-5.4-c1-0-21
Degree $2$
Conductor $1815$
Sign $-0.994 + 0.100i$
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  + 1.41i·2-s i·3-s + (0.224 + 2.22i)5-s + 1.41·6-s + 1.73i·7-s + 2.82i·8-s − 9-s + (−3.14 + 0.317i)10-s + 6.29i·13-s − 2.44·14-s + (2.22 − 0.224i)15-s − 4.00·16-s − 2.04i·17-s − 1.41i·18-s − 1.73·19-s + ⋯
L(s)  = 1  + 0.999i·2-s − 0.577i·3-s + (0.100 + 0.994i)5-s + 0.577·6-s + 0.654i·7-s + 0.999i·8-s − 0.333·9-s + (−0.994 + 0.100i)10-s + 1.74i·13-s − 0.654·14-s + (0.574 − 0.0580i)15-s − 1.00·16-s − 0.497i·17-s − 0.333i·18-s − 0.397·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.100i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 + 0.100i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1815\)    =    \(3 \cdot 5 \cdot 11^{2}\)
Sign: $-0.994 + 0.100i$
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: \((2,\ 1815,\ (\ :1/2),\ -0.994 + 0.100i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.478735907\)
\(L(\frac12)\) \(\approx\) \(1.478735907\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.224 - 2.22i)T \)
11 \( 1 \)
good2 \( 1 - 1.41iT - 2T^{2} \)
7 \( 1 - 1.73iT - 7T^{2} \)
13 \( 1 - 6.29iT - 13T^{2} \)
17 \( 1 + 2.04iT - 17T^{2} \)
19 \( 1 + 1.73T + 19T^{2} \)
23 \( 1 + 2iT - 23T^{2} \)
29 \( 1 + 5.51T + 29T^{2} \)
31 \( 1 - 1.89T + 31T^{2} \)
37 \( 1 + 3.89iT - 37T^{2} \)
41 \( 1 - 2.19T + 41T^{2} \)
43 \( 1 + 5.65iT - 43T^{2} \)
47 \( 1 + 7.34iT - 47T^{2} \)
53 \( 1 - 10iT - 53T^{2} \)
59 \( 1 + 12.2T + 59T^{2} \)
61 \( 1 - 13.6T + 61T^{2} \)
67 \( 1 + 13.8iT - 67T^{2} \)
71 \( 1 + 8.44T + 71T^{2} \)
73 \( 1 - 13.6iT - 73T^{2} \)
79 \( 1 - 10.2T + 79T^{2} \)
83 \( 1 - 12.7iT - 83T^{2} \)
89 \( 1 - 13.3T + 89T^{2} \)
97 \( 1 + 5iT - 97T^{2} \)
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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

−9.293827209035269354146894733404, −8.812337400457492465503066093197, −7.77628116230317201170794076396, −7.15661657989102346569605415551, −6.57413895663002701794098046152, −6.02660871178717852995381432162, −5.14300436615255347770604853053, −3.90402935671172400495731823744, −2.50427074126901049129634797628, −2.01013625963307612993191875221, 0.51457974833832237274760943081, 1.59048223297747150126521059862, 2.89011194626489841263249353849, 3.72528023738358523613633165705, 4.48643402222868496093574621170, 5.45243376895937187689253435576, 6.26237075056535629437133461072, 7.53386919001246126719004635206, 8.167834089258285527517110769344, 9.130041338184679436583121388538

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