Properties

Label 2-214774-1.1-c1-0-8
Degree $2$
Conductor $214774$
Sign $-1$
Analytic cond. $1714.97$
Root an. cond. $41.4123$
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 − 2·5-s + 7-s + 8-s − 3·9-s − 2·10-s − 4·11-s + 6·13-s + 14-s + 16-s − 6·17-s − 3·18-s − 4·19-s − 2·20-s − 4·22-s − 25-s + 6·26-s + 28-s + 29-s − 4·31-s + 32-s − 6·34-s − 2·35-s − 3·36-s − 6·37-s − 4·38-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.894·5-s + 0.377·7-s + 0.353·8-s − 9-s − 0.632·10-s − 1.20·11-s + 1.66·13-s + 0.267·14-s + 1/4·16-s − 1.45·17-s − 0.707·18-s − 0.917·19-s − 0.447·20-s − 0.852·22-s − 1/5·25-s + 1.17·26-s + 0.188·28-s + 0.185·29-s − 0.718·31-s + 0.176·32-s − 1.02·34-s − 0.338·35-s − 1/2·36-s − 0.986·37-s − 0.648·38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 214774 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 214774 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(214774\)    =    \(2 \cdot 7 \cdot 23^{2} \cdot 29\)
Sign: $-1$
Analytic conductor: \(1714.97\)
Root analytic conductor: \(41.4123\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{214774} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 214774,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
7 \( 1 - T \)
23 \( 1 \)
29 \( 1 - T \)
good3 \( 1 + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−13.25458463158386, −12.73202243470962, −12.45324429647209, −11.60694216976558, −11.48285023868995, −10.93161316288953, −10.70079065405408, −10.32446435011027, −9.224600819988631, −8.916387465084665, −8.293205117518331, −8.127270521262668, −7.641980225299490, −6.790543647755397, −6.589300811405333, −5.857408071407653, −5.531294004203871, −4.888722462110010, −4.437199924474199, −3.814210740352360, −3.481916751567585, −2.846516324449024, −2.167062736944059, −1.771828769893423, −0.6486846023569878, 0, 0.6486846023569878, 1.771828769893423, 2.167062736944059, 2.846516324449024, 3.481916751567585, 3.814210740352360, 4.437199924474199, 4.888722462110010, 5.531294004203871, 5.857408071407653, 6.589300811405333, 6.790543647755397, 7.641980225299490, 8.127270521262668, 8.293205117518331, 8.916387465084665, 9.224600819988631, 10.32446435011027, 10.70079065405408, 10.93161316288953, 11.48285023868995, 11.60694216976558, 12.45324429647209, 12.73202243470962, 13.25458463158386

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