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Properties

Label 2-34650-1.1-c1-0-112
Degree $2$
Conductor $34650$
Sign $-1$
Analytic cond. $276.681$
Root an. cond. $16.6337$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 7-s + 8-s + 11-s − 2·13-s + 14-s + 16-s + 2·17-s + 4·19-s + 22-s − 4·23-s − 2·26-s + 28-s + 6·29-s + 32-s + 2·34-s − 2·37-s + 4·38-s − 6·41-s − 12·43-s + 44-s − 4·46-s + 8·47-s + 49-s − 2·52-s − 6·53-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.377·7-s + 0.353·8-s + 0.301·11-s − 0.554·13-s + 0.267·14-s + 1/4·16-s + 0.485·17-s + 0.917·19-s + 0.213·22-s − 0.834·23-s − 0.392·26-s + 0.188·28-s + 1.11·29-s + 0.176·32-s + 0.342·34-s − 0.328·37-s + 0.648·38-s − 0.937·41-s − 1.82·43-s + 0.150·44-s − 0.589·46-s + 1.16·47-s + 1/7·49-s − 0.277·52-s − 0.824·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 34650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 34650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(34650\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \cdot 11\)
Sign: $-1$
Analytic conductor: \(276.681\)
Root analytic conductor: \(16.6337\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 34650,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
7 \( 1 - T \)
11 \( 1 - T \)
good13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−15.07517328909267, −14.75242356000703, −13.95838621801013, −13.82347743091918, −13.31476983577811, −12.34342188471660, −12.12031185824167, −11.79973838834993, −11.10801563493787, −10.34095719368636, −10.14336466495494, −9.348699450429109, −8.787991480476463, −7.985717541882105, −7.660420633103088, −6.980023740147829, −6.378521261657763, −5.803737757666094, −5.125084534690469, −4.684429230704452, −4.048922670258052, −3.211519393269695, −2.843459706345583, −1.793432636634637, −1.294023965998709, 0, 1.294023965998709, 1.793432636634637, 2.843459706345583, 3.211519393269695, 4.048922670258052, 4.684429230704452, 5.125084534690469, 5.803737757666094, 6.378521261657763, 6.980023740147829, 7.660420633103088, 7.985717541882105, 8.787991480476463, 9.348699450429109, 10.14336466495494, 10.34095719368636, 11.10801563493787, 11.79973838834993, 12.12031185824167, 12.34342188471660, 13.31476983577811, 13.82347743091918, 13.95838621801013, 14.75242356000703, 15.07517328909267

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