Properties

Degree $2$
Conductor $102$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 4-s − 2·5-s + 6-s + 8-s + 9-s − 2·10-s − 4·11-s + 12-s − 2·13-s − 2·15-s + 16-s + 17-s + 18-s + 4·19-s − 2·20-s − 4·22-s + 24-s − 25-s − 2·26-s + 27-s − 10·29-s − 2·30-s + 8·31-s + 32-s − 4·33-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.894·5-s + 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.632·10-s − 1.20·11-s + 0.288·12-s − 0.554·13-s − 0.516·15-s + 1/4·16-s + 0.242·17-s + 0.235·18-s + 0.917·19-s − 0.447·20-s − 0.852·22-s + 0.204·24-s − 1/5·25-s − 0.392·26-s + 0.192·27-s − 1.85·29-s − 0.365·30-s + 1.43·31-s + 0.176·32-s − 0.696·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 102 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(102\)    =    \(2 \cdot 3 \cdot 17\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{102} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 102,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.47967\)
\(L(\frac12)\) \(\approx\) \(1.47967\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 - T \)
17 \( 1 - T \)
good5 \( 1 + 2 T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−13.82453782976520878118936880008, −12.87130622556280185790540229129, −11.93324918288479588422006823319, −10.84353554600873070196710460127, −9.570394810938145814587700544156, −7.958882066985845549410362974675, −7.36440720073199929079936076774, −5.51457882926074302977186886250, −4.13875090005906653574798019221, −2.76309813409348720224135008444, 2.76309813409348720224135008444, 4.13875090005906653574798019221, 5.51457882926074302977186886250, 7.36440720073199929079936076774, 7.958882066985845549410362974675, 9.570394810938145814587700544156, 10.84353554600873070196710460127, 11.93324918288479588422006823319, 12.87130622556280185790540229129, 13.82453782976520878118936880008

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