Properties

Label 2-912-1.1-c3-0-11
Degree $2$
Conductor $912$
Sign $1$
Analytic cond. $53.8097$
Root an. cond. $7.33551$
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  − 3·3-s − 7·5-s + 15·7-s + 9·9-s + 49·11-s + 14·13-s + 21·15-s − 33·17-s + 19·19-s − 45·21-s + 148·23-s − 76·25-s − 27·27-s − 278·29-s − 94·31-s − 147·33-s − 105·35-s + 160·37-s − 42·39-s + 400·41-s − 73·43-s − 63·45-s − 173·47-s − 118·49-s + 99·51-s + 170·53-s − 343·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.626·5-s + 0.809·7-s + 1/3·9-s + 1.34·11-s + 0.298·13-s + 0.361·15-s − 0.470·17-s + 0.229·19-s − 0.467·21-s + 1.34·23-s − 0.607·25-s − 0.192·27-s − 1.78·29-s − 0.544·31-s − 0.775·33-s − 0.507·35-s + 0.710·37-s − 0.172·39-s + 1.52·41-s − 0.258·43-s − 0.208·45-s − 0.536·47-s − 0.344·49-s + 0.271·51-s + 0.440·53-s − 0.840·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $1$
Analytic conductor: \(53.8097\)
Root analytic conductor: \(7.33551\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: $\chi_{912} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.739924905\)
\(L(\frac12)\) \(\approx\) \(1.739924905\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + p T \)
19 \( 1 - p T \)
good5 \( 1 + 7 T + p^{3} T^{2} \)
7 \( 1 - 15 T + p^{3} T^{2} \)
11 \( 1 - 49 T + p^{3} T^{2} \)
13 \( 1 - 14 T + p^{3} T^{2} \)
17 \( 1 + 33 T + p^{3} T^{2} \)
23 \( 1 - 148 T + p^{3} T^{2} \)
29 \( 1 + 278 T + p^{3} T^{2} \)
31 \( 1 + 94 T + p^{3} T^{2} \)
37 \( 1 - 160 T + p^{3} T^{2} \)
41 \( 1 - 400 T + p^{3} T^{2} \)
43 \( 1 + 73 T + p^{3} T^{2} \)
47 \( 1 + 173 T + p^{3} T^{2} \)
53 \( 1 - 170 T + p^{3} T^{2} \)
59 \( 1 - 12 T + p^{3} T^{2} \)
61 \( 1 - 419 T + p^{3} T^{2} \)
67 \( 1 + 444 T + p^{3} T^{2} \)
71 \( 1 - 952 T + p^{3} T^{2} \)
73 \( 1 + 27 T + p^{3} T^{2} \)
79 \( 1 - 556 T + p^{3} T^{2} \)
83 \( 1 - 276 T + p^{3} T^{2} \)
89 \( 1 - 1386 T + p^{3} T^{2} \)
97 \( 1 - 130 T + p^{3} T^{2} \)
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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

−9.556392717028457585239498153050, −8.973340666111262649883877927079, −7.904685113893163395493555917809, −7.18490926444383802894774417904, −6.27754130620658587456707528300, −5.29311764947051825021823018017, −4.32533145980041930446190233461, −3.58427192136092151025726774775, −1.87355512482263125898090653131, −0.77043222100486511123292230306, 0.77043222100486511123292230306, 1.87355512482263125898090653131, 3.58427192136092151025726774775, 4.32533145980041930446190233461, 5.29311764947051825021823018017, 6.27754130620658587456707528300, 7.18490926444383802894774417904, 7.904685113893163395493555917809, 8.973340666111262649883877927079, 9.556392717028457585239498153050

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