Properties

Degree $2$
Conductor $2015$
Sign $1$
Motivic weight $0$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.94·2-s + 1.49·3-s + 2.77·4-s + 5-s − 2.90·6-s + 1.13·7-s − 3.43·8-s + 1.24·9-s − 1.94·10-s + 0.709·11-s + 4.14·12-s − 13-s − 2.20·14-s + 1.49·15-s + 3.90·16-s − 1.77·17-s − 2.41·18-s + 2.77·20-s + 1.70·21-s − 1.37·22-s − 0.241·23-s − 5.14·24-s + 25-s + 1.94·26-s + 0.360·27-s + 3.14·28-s − 2.90·30-s + ⋯
L(s)  = 1  − 1.94·2-s + 1.49·3-s + 2.77·4-s + 5-s − 2.90·6-s + 1.13·7-s − 3.43·8-s + 1.24·9-s − 1.94·10-s + 0.709·11-s + 4.14·12-s − 13-s − 2.20·14-s + 1.49·15-s + 3.90·16-s − 1.77·17-s − 2.41·18-s + 2.77·20-s + 1.70·21-s − 1.37·22-s − 0.241·23-s − 5.14·24-s + 25-s + 1.94·26-s + 0.360·27-s + 3.14·28-s − 2.90·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2015\)    =    \(5 \cdot 13 \cdot 31\)
Sign: $1$
Motivic weight: \(0\)
Character: $\chi_{2015} (2014, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2015,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.075658375\)
\(L(\frac12)\) \(\approx\) \(1.075658375\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - T \)
13 \( 1 + T \)
31 \( 1 + T \)
good2 \( 1 + 1.94T + T^{2} \)
3 \( 1 - 1.49T + T^{2} \)
7 \( 1 - 1.13T + T^{2} \)
11 \( 1 - 0.709T + T^{2} \)
17 \( 1 + 1.77T + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + 0.241T + T^{2} \)
29 \( 1 - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - 1.94T + T^{2} \)
47 \( 1 - 0.241T + T^{2} \)
53 \( 1 - 0.709T + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + 0.709T + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + 1.13T + T^{2} \)
97 \( 1 + 1.49T + T^{2} \)
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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

−9.264748479643630675920452214235, −8.725251287021768837704294158751, −8.107364763971308176035140655065, −7.30492846112170650436488710201, −6.78209452829725625667304858587, −5.65513457445039859392847768437, −4.23955465756395753043709103724, −2.70983948529623984331796936912, −2.17478861680126184669907456665, −1.51671490132805965991072806610, 1.51671490132805965991072806610, 2.17478861680126184669907456665, 2.70983948529623984331796936912, 4.23955465756395753043709103724, 5.65513457445039859392847768437, 6.78209452829725625667304858587, 7.30492846112170650436488710201, 8.107364763971308176035140655065, 8.725251287021768837704294158751, 9.264748479643630675920452214235

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