Properties

Label 2-4025-1.1-c1-0-168
Degree $2$
Conductor $4025$
Sign $1$
Analytic cond. $32.1397$
Root an. cond. $5.66919$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.15·2-s + 2.62·3-s + 2.64·4-s + 5.65·6-s + 7-s + 1.39·8-s + 3.88·9-s + 3.15·11-s + 6.94·12-s + 1.72·13-s + 2.15·14-s − 2.28·16-s + 5.38·17-s + 8.37·18-s − 3.42·19-s + 2.62·21-s + 6.80·22-s − 23-s + 3.65·24-s + 3.72·26-s + 2.32·27-s + 2.64·28-s − 2.54·29-s − 9.96·31-s − 7.72·32-s + 8.28·33-s + 11.6·34-s + ⋯
L(s)  = 1  + 1.52·2-s + 1.51·3-s + 1.32·4-s + 2.30·6-s + 0.377·7-s + 0.492·8-s + 1.29·9-s + 0.951·11-s + 2.00·12-s + 0.479·13-s + 0.576·14-s − 0.572·16-s + 1.30·17-s + 1.97·18-s − 0.786·19-s + 0.572·21-s + 1.45·22-s − 0.208·23-s + 0.746·24-s + 0.731·26-s + 0.446·27-s + 0.500·28-s − 0.471·29-s − 1.79·31-s − 1.36·32-s + 1.44·33-s + 1.99·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4025\)    =    \(5^{2} \cdot 7 \cdot 23\)
Sign: $1$
Analytic conductor: \(32.1397\)
Root analytic conductor: \(5.66919\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4025,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(8.506944478\)
\(L(\frac12)\) \(\approx\) \(8.506944478\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 - T \)
23 \( 1 + T \)
good2 \( 1 - 2.15T + 2T^{2} \)
3 \( 1 - 2.62T + 3T^{2} \)
11 \( 1 - 3.15T + 11T^{2} \)
13 \( 1 - 1.72T + 13T^{2} \)
17 \( 1 - 5.38T + 17T^{2} \)
19 \( 1 + 3.42T + 19T^{2} \)
29 \( 1 + 2.54T + 29T^{2} \)
31 \( 1 + 9.96T + 31T^{2} \)
37 \( 1 - 5.69T + 37T^{2} \)
41 \( 1 - 7.30T + 41T^{2} \)
43 \( 1 + 4.93T + 43T^{2} \)
47 \( 1 - 7.42T + 47T^{2} \)
53 \( 1 - 3.20T + 53T^{2} \)
59 \( 1 - 8.27T + 59T^{2} \)
61 \( 1 + 7.95T + 61T^{2} \)
67 \( 1 + 5.20T + 67T^{2} \)
71 \( 1 - 13.0T + 71T^{2} \)
73 \( 1 + 8.81T + 73T^{2} \)
79 \( 1 + 4.05T + 79T^{2} \)
83 \( 1 + 12.0T + 83T^{2} \)
89 \( 1 - 5.22T + 89T^{2} \)
97 \( 1 + 4.36T + 97T^{2} \)
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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

−8.412270440098431439194727290302, −7.61942149001274668470954684709, −6.98974067022530953276596287167, −6.00266084422065133348796751439, −5.44024190877397346366028629487, −4.22701204152722738470172863806, −3.91420671998509982133218308342, −3.20798793580152778132755992475, −2.35772052916847044614722097571, −1.49149066884146795578000399191, 1.49149066884146795578000399191, 2.35772052916847044614722097571, 3.20798793580152778132755992475, 3.91420671998509982133218308342, 4.22701204152722738470172863806, 5.44024190877397346366028629487, 6.00266084422065133348796751439, 6.98974067022530953276596287167, 7.61942149001274668470954684709, 8.412270440098431439194727290302

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