Properties

Label 2-1815-1.1-c1-0-15
Degree $2$
Conductor $1815$
Sign $1$
Analytic cond. $14.4928$
Root an. cond. $3.80694$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.63·2-s − 3-s + 4.92·4-s + 5-s + 2.63·6-s + 4.16·7-s − 7.69·8-s + 9-s − 2.63·10-s − 4.92·12-s − 5.26·13-s − 10.9·14-s − 15-s + 10.4·16-s − 4.23·17-s − 2.63·18-s + 2.50·19-s + 4.92·20-s − 4.16·21-s + 5.48·23-s + 7.69·24-s + 25-s + 13.8·26-s − 27-s + 20.5·28-s − 5.26·29-s + 2.63·30-s + ⋯
L(s)  = 1  − 1.86·2-s − 0.577·3-s + 2.46·4-s + 0.447·5-s + 1.07·6-s + 1.57·7-s − 2.72·8-s + 0.333·9-s − 0.832·10-s − 1.42·12-s − 1.45·13-s − 2.93·14-s − 0.258·15-s + 2.60·16-s − 1.02·17-s − 0.620·18-s + 0.574·19-s + 1.10·20-s − 0.909·21-s + 1.14·23-s + 1.57·24-s + 0.200·25-s + 2.71·26-s − 0.192·27-s + 3.87·28-s − 0.977·29-s + 0.480·30-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1815\)    =    \(3 \cdot 5 \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(14.4928\)
Root analytic conductor: \(3.80694\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1815} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1815,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7253878749\)
\(L(\frac12)\) \(\approx\) \(0.7253878749\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
5 \( 1 - T \)
11 \( 1 \)
good2 \( 1 + 2.63T + 2T^{2} \)
7 \( 1 - 4.16T + 7T^{2} \)
13 \( 1 + 5.26T + 13T^{2} \)
17 \( 1 + 4.23T + 17T^{2} \)
19 \( 1 - 2.50T + 19T^{2} \)
23 \( 1 - 5.48T + 23T^{2} \)
29 \( 1 + 5.26T + 29T^{2} \)
31 \( 1 - 10.1T + 31T^{2} \)
37 \( 1 + 4.48T + 37T^{2} \)
41 \( 1 - 3.46T + 41T^{2} \)
43 \( 1 - 6.66T + 43T^{2} \)
47 \( 1 - 4.21T + 47T^{2} \)
53 \( 1 + 3.63T + 53T^{2} \)
59 \( 1 + 6.58T + 59T^{2} \)
61 \( 1 - 12.2T + 61T^{2} \)
67 \( 1 + 4.48T + 67T^{2} \)
71 \( 1 - 1.26T + 71T^{2} \)
73 \( 1 + 4.55T + 73T^{2} \)
79 \( 1 - 6.86T + 79T^{2} \)
83 \( 1 - 4.87T + 83T^{2} \)
89 \( 1 - 15.1T + 89T^{2} \)
97 \( 1 + 16.1T + 97T^{2} \)
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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

−9.245989143023734165906564912397, −8.608121810429003359026359621248, −7.67448216439699323302855970509, −7.27770942938493002582391848150, −6.40710560035774514124569498517, −5.32630889446843402776215303962, −4.60126583134952915227422915190, −2.62591865621093436354674313949, −1.86176657611354719519191783647, −0.803600380337510547049285616570, 0.803600380337510547049285616570, 1.86176657611354719519191783647, 2.62591865621093436354674313949, 4.60126583134952915227422915190, 5.32630889446843402776215303962, 6.40710560035774514124569498517, 7.27770942938493002582391848150, 7.67448216439699323302855970509, 8.608121810429003359026359621248, 9.245989143023734165906564912397

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