Properties

Label 2-15-1.1-c3-0-0
Degree $2$
Conductor $15$
Sign $1$
Analytic cond. $0.885028$
Root an. cond. $0.940759$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3·3-s − 7·4-s + 5·5-s + 3·6-s − 24·7-s − 15·8-s + 9·9-s + 5·10-s + 52·11-s − 21·12-s + 22·13-s − 24·14-s + 15·15-s + 41·16-s − 14·17-s + 9·18-s − 20·19-s − 35·20-s − 72·21-s + 52·22-s − 168·23-s − 45·24-s + 25·25-s + 22·26-s + 27·27-s + 168·28-s + ⋯
L(s)  = 1  + 0.353·2-s + 0.577·3-s − 7/8·4-s + 0.447·5-s + 0.204·6-s − 1.29·7-s − 0.662·8-s + 1/3·9-s + 0.158·10-s + 1.42·11-s − 0.505·12-s + 0.469·13-s − 0.458·14-s + 0.258·15-s + 0.640·16-s − 0.199·17-s + 0.117·18-s − 0.241·19-s − 0.391·20-s − 0.748·21-s + 0.503·22-s − 1.52·23-s − 0.382·24-s + 1/5·25-s + 0.165·26-s + 0.192·27-s + 1.13·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $1$
Analytic conductor: \(0.885028\)
Root analytic conductor: \(0.940759\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: $\chi_{15} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 15,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.139433421\)
\(L(\frac12)\) \(\approx\) \(1.139433421\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - p T \)
5 \( 1 - p T \)
good2 \( 1 - T + p^{3} T^{2} \)
7 \( 1 + 24 T + p^{3} T^{2} \)
11 \( 1 - 52 T + p^{3} T^{2} \)
13 \( 1 - 22 T + p^{3} T^{2} \)
17 \( 1 + 14 T + p^{3} T^{2} \)
19 \( 1 + 20 T + p^{3} T^{2} \)
23 \( 1 + 168 T + p^{3} T^{2} \)
29 \( 1 - 230 T + p^{3} T^{2} \)
31 \( 1 + 288 T + p^{3} T^{2} \)
37 \( 1 + 34 T + p^{3} T^{2} \)
41 \( 1 - 122 T + p^{3} T^{2} \)
43 \( 1 + 188 T + p^{3} T^{2} \)
47 \( 1 - 256 T + p^{3} T^{2} \)
53 \( 1 + 338 T + p^{3} T^{2} \)
59 \( 1 - 100 T + p^{3} T^{2} \)
61 \( 1 - 742 T + p^{3} T^{2} \)
67 \( 1 + 84 T + p^{3} T^{2} \)
71 \( 1 + 328 T + p^{3} T^{2} \)
73 \( 1 + 38 T + p^{3} T^{2} \)
79 \( 1 + 240 T + p^{3} T^{2} \)
83 \( 1 - 1212 T + p^{3} T^{2} \)
89 \( 1 - 330 T + p^{3} T^{2} \)
97 \( 1 - 866 T + p^{3} T^{2} \)
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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

−19.02881405917150528687290867331, −17.70949836886805470548052216760, −16.19372858254051139851659443609, −14.50847812106069670742108136410, −13.56121835041038960562334260750, −12.39876426593880772586471717594, −9.853422404707595692052539772777, −8.847113744022096962511239249312, −6.30445355770960208718872883735, −3.78175129911665234242031093429, 3.78175129911665234242031093429, 6.30445355770960208718872883735, 8.847113744022096962511239249312, 9.853422404707595692052539772777, 12.39876426593880772586471717594, 13.56121835041038960562334260750, 14.50847812106069670742108136410, 16.19372858254051139851659443609, 17.70949836886805470548052216760, 19.02881405917150528687290867331

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