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Properties

Label 2-25451-1.1-c1-0-2
Degree $2$
Conductor $25451$
Sign $-1$
Analytic cond. $203.227$
Root an. cond. $14.2557$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $3$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s − 4-s − 2·5-s + 6-s − 4·7-s + 3·8-s − 2·9-s + 2·10-s − 3·11-s + 12-s − 2·13-s + 4·14-s + 2·15-s − 16-s − 8·17-s + 2·18-s − 8·19-s + 2·20-s + 4·21-s + 3·22-s − 6·23-s − 3·24-s − 25-s + 2·26-s + 5·27-s + 4·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.894·5-s + 0.408·6-s − 1.51·7-s + 1.06·8-s − 2/3·9-s + 0.632·10-s − 0.904·11-s + 0.288·12-s − 0.554·13-s + 1.06·14-s + 0.516·15-s − 1/4·16-s − 1.94·17-s + 0.471·18-s − 1.83·19-s + 0.447·20-s + 0.872·21-s + 0.639·22-s − 1.25·23-s − 0.612·24-s − 1/5·25-s + 0.392·26-s + 0.962·27-s + 0.755·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25451 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25451 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(25451\)    =    \(31 \cdot 821\)
Sign: $-1$
Analytic conductor: \(203.227\)
Root analytic conductor: \(14.2557\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((2,\ 25451,\ (\ :1/2),\ -1)\)

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)$
bad31 \( 1 + T \)
821 \( 1 + T \)
good2 \( 1 + T + p T^{2} \)
3 \( 1 + T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 8 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
37 \( 1 + 5 T + p T^{2} \)
41 \( 1 + 5 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 7 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 5 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 9 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 3 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−16.21099773359544, −15.70739898874568, −15.32833442747625, −14.68939717355943, −13.85452300872749, −13.22083629817289, −13.07010574133788, −12.31669612616459, −11.89339460257378, −11.07858703086477, −10.65261040503920, −10.25141175520155, −9.579535390628767, −8.981206392924087, −8.454458963836604, −8.067480315051337, −7.282630219835071, −6.630742806245797, −6.232753600212845, −5.410711645374236, −4.673815648283637, −4.147442523228151, −3.511909650891008, −2.591454147710290, −1.897767667070918, 0, 0, 0, 1.897767667070918, 2.591454147710290, 3.511909650891008, 4.147442523228151, 4.673815648283637, 5.410711645374236, 6.232753600212845, 6.630742806245797, 7.282630219835071, 8.067480315051337, 8.454458963836604, 8.981206392924087, 9.579535390628767, 10.25141175520155, 10.65261040503920, 11.07858703086477, 11.89339460257378, 12.31669612616459, 13.07010574133788, 13.22083629817289, 13.85452300872749, 14.68939717355943, 15.32833442747625, 15.70739898874568, 16.21099773359544

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