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Properties

Label 2-1815-5.4-c1-0-37
Degree $2$
Conductor $1815$
Sign $0.164 - 0.986i$
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  − 0.0838i·2-s + i·3-s + 1.99·4-s + (−2.20 − 0.366i)5-s + 0.0838·6-s + 2.34i·7-s − 0.334i·8-s − 9-s + (−0.0307 + 0.184i)10-s + 1.99i·12-s − 5.61i·13-s + 0.196·14-s + (0.366 − 2.20i)15-s + 3.95·16-s + 2.13i·17-s + 0.0838i·18-s + ⋯
L(s)  = 1  − 0.0592i·2-s + 0.577i·3-s + 0.996·4-s + (−0.986 − 0.164i)5-s + 0.0342·6-s + 0.885i·7-s − 0.118i·8-s − 0.333·9-s + (−0.00972 + 0.0584i)10-s + 0.575i·12-s − 1.55i·13-s + 0.0524·14-s + (0.0947 − 0.569i)15-s + 0.989·16-s + 0.518i·17-s + 0.0197i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.164 - 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.164 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1815\)    =    \(3 \cdot 5 \cdot 11^{2}\)
Sign: $0.164 - 0.986i$
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: \((2,\ 1815,\ (\ :1/2),\ 0.164 - 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.718918737\)
\(L(\frac12)\) \(\approx\) \(1.718918737\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.20 + 0.366i)T \)
11 \( 1 \)
good2 \( 1 + 0.0838iT - 2T^{2} \)
7 \( 1 - 2.34iT - 7T^{2} \)
13 \( 1 + 5.61iT - 13T^{2} \)
17 \( 1 - 2.13iT - 17T^{2} \)
19 \( 1 - 2.17T + 19T^{2} \)
23 \( 1 - 8.11iT - 23T^{2} \)
29 \( 1 - 5.11T + 29T^{2} \)
31 \( 1 - 4.93T + 31T^{2} \)
37 \( 1 - 5.25iT - 37T^{2} \)
41 \( 1 + 8.59T + 41T^{2} \)
43 \( 1 - 6.99iT - 43T^{2} \)
47 \( 1 - 5.75iT - 47T^{2} \)
53 \( 1 - 10.3iT - 53T^{2} \)
59 \( 1 - 3.46T + 59T^{2} \)
61 \( 1 + 11.7T + 61T^{2} \)
67 \( 1 - 9.16iT - 67T^{2} \)
71 \( 1 + 6.51T + 71T^{2} \)
73 \( 1 + 13.7iT - 73T^{2} \)
79 \( 1 + 6.68T + 79T^{2} \)
83 \( 1 - 11.7iT - 83T^{2} \)
89 \( 1 + 2.38T + 89T^{2} \)
97 \( 1 + 2.10iT - 97T^{2} \)
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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

−9.532268246166361598645705889307, −8.432863127504887215088957390852, −7.989977829274183760142183268565, −7.21648640485392602974940449528, −6.07365092173712186456462123701, −5.48914911751669499533011260939, −4.51431623707400883298664213586, −3.16570468780321897398580502383, −2.99351219985926007555030120879, −1.28413478539086750149541925742, 0.67892676202920843920144243663, 2.00228663934252416442980908631, 3.03753808817187583717679273747, 4.03248412247338406483603555749, 4.89409163808406055568201646985, 6.31323963224054258160499930419, 6.95462815817528725573529763582, 7.19313868915962873126361506272, 8.166641167600763441613174767294, 8.809268637345850074363249383748

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