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Properties

Label 2-214774-1.1-c1-0-15
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 $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 2·3-s + 4-s − 2·6-s + 7-s + 8-s + 9-s − 4·11-s − 2·12-s − 2·13-s + 14-s + 16-s − 4·17-s + 18-s − 2·21-s − 4·22-s − 2·24-s − 5·25-s − 2·26-s + 4·27-s + 28-s − 29-s − 10·31-s + 32-s + 8·33-s − 4·34-s + 36-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.816·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 1.20·11-s − 0.577·12-s − 0.554·13-s + 0.267·14-s + 1/4·16-s − 0.970·17-s + 0.235·18-s − 0.436·21-s − 0.852·22-s − 0.408·24-s − 25-s − 0.392·26-s + 0.769·27-s + 0.188·28-s − 0.185·29-s − 1.79·31-s + 0.176·32-s + 1.39·33-s − 0.685·34-s + 1/6·36-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
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 \( 1 + 2 T + p T^{2} \)
5 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 + 16 T + p T^{2} \)
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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

−13.45489197568038, −12.86638848751666, −12.51788513396896, −12.09573209500721, −11.62866765307561, −11.05306319402900, −10.86209100041023, −10.54868587195307, −9.803638615117500, −9.372794943869456, −8.724101211852286, −8.072366672913293, −7.620112671313212, −7.246061317945923, −6.553446724220298, −6.195940132579799, −5.574779802097466, −5.248658033921542, −4.874349050024385, −4.348600548976936, −3.720256770327445, −3.081963504980266, −2.310037063565695, −2.030160670563166, −1.146341410095603, 0, 0, 1.146341410095603, 2.030160670563166, 2.310037063565695, 3.081963504980266, 3.720256770327445, 4.348600548976936, 4.874349050024385, 5.248658033921542, 5.574779802097466, 6.195940132579799, 6.553446724220298, 7.246061317945923, 7.620112671313212, 8.072366672913293, 8.724101211852286, 9.372794943869456, 9.803638615117500, 10.54868587195307, 10.86209100041023, 11.05306319402900, 11.62866765307561, 12.09573209500721, 12.51788513396896, 12.86638848751666, 13.45489197568038

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