Properties

Label 2-1815-1.1-c3-0-35
Degree $2$
Conductor $1815$
Sign $1$
Analytic cond. $107.088$
Root an. cond. $10.3483$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.97·2-s + 3·3-s + 16.7·4-s − 5·5-s − 14.9·6-s − 5.48·7-s − 43.3·8-s + 9·9-s + 24.8·10-s + 50.1·12-s − 24.5·13-s + 27.2·14-s − 15·15-s + 81.6·16-s + 59.3·17-s − 44.7·18-s − 5.89·19-s − 83.5·20-s − 16.4·21-s − 68.4·23-s − 129.·24-s + 25·25-s + 122.·26-s + 27·27-s − 91.6·28-s + 265.·29-s + 74.5·30-s + ⋯
L(s)  = 1  − 1.75·2-s + 0.577·3-s + 2.08·4-s − 0.447·5-s − 1.01·6-s − 0.296·7-s − 1.91·8-s + 0.333·9-s + 0.786·10-s + 1.20·12-s − 0.524·13-s + 0.520·14-s − 0.258·15-s + 1.27·16-s + 0.847·17-s − 0.585·18-s − 0.0711·19-s − 0.934·20-s − 0.170·21-s − 0.620·23-s − 1.10·24-s + 0.200·25-s + 0.921·26-s + 0.192·27-s − 0.618·28-s + 1.69·29-s + 0.453·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1815\)    =    \(3 \cdot 5 \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(107.088\)
Root analytic conductor: \(10.3483\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1815} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1815,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.7812256604\)
\(L(\frac12)\) \(\approx\) \(0.7812256604\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 3T \)
5 \( 1 + 5T \)
11 \( 1 \)
good2 \( 1 + 4.97T + 8T^{2} \)
7 \( 1 + 5.48T + 343T^{2} \)
13 \( 1 + 24.5T + 2.19e3T^{2} \)
17 \( 1 - 59.3T + 4.91e3T^{2} \)
19 \( 1 + 5.89T + 6.85e3T^{2} \)
23 \( 1 + 68.4T + 1.21e4T^{2} \)
29 \( 1 - 265.T + 2.43e4T^{2} \)
31 \( 1 + 196.T + 2.97e4T^{2} \)
37 \( 1 - 166.T + 5.06e4T^{2} \)
41 \( 1 + 424.T + 6.89e4T^{2} \)
43 \( 1 - 177.T + 7.95e4T^{2} \)
47 \( 1 - 141.T + 1.03e5T^{2} \)
53 \( 1 + 339.T + 1.48e5T^{2} \)
59 \( 1 - 416.T + 2.05e5T^{2} \)
61 \( 1 + 662.T + 2.26e5T^{2} \)
67 \( 1 - 313.T + 3.00e5T^{2} \)
71 \( 1 + 153.T + 3.57e5T^{2} \)
73 \( 1 + 153.T + 3.89e5T^{2} \)
79 \( 1 + 403.T + 4.93e5T^{2} \)
83 \( 1 - 652.T + 5.71e5T^{2} \)
89 \( 1 - 1.22e3T + 7.04e5T^{2} \)
97 \( 1 - 959.T + 9.12e5T^{2} \)
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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

−8.901677878101483164732545978261, −8.146845535110716473013041916255, −7.68155212738953520809810581499, −6.93837220696184295005558493422, −6.13353153738808215739894937211, −4.78658643954611606608651611457, −3.50939886386724847681186909810, −2.64072958350920720113026199117, −1.61778701123934422766480284896, −0.53196175745247068709289621720, 0.53196175745247068709289621720, 1.61778701123934422766480284896, 2.64072958350920720113026199117, 3.50939886386724847681186909810, 4.78658643954611606608651611457, 6.13353153738808215739894937211, 6.93837220696184295005558493422, 7.68155212738953520809810581499, 8.146845535110716473013041916255, 8.901677878101483164732545978261

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