Properties

Label 2-3174-1.1-c1-0-35
Degree $2$
Conductor $3174$
Sign $1$
Analytic cond. $25.3445$
Root an. cond. $5.03433$
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-s − 3-s + 4-s − 0.194·5-s − 6-s + 1.95·7-s + 8-s + 9-s − 0.194·10-s + 6.30·11-s − 12-s + 4.81·13-s + 1.95·14-s + 0.194·15-s + 16-s − 3.20·17-s + 18-s + 0.859·19-s − 0.194·20-s − 1.95·21-s + 6.30·22-s − 24-s − 4.96·25-s + 4.81·26-s − 27-s + 1.95·28-s + 5.41·29-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.0868·5-s − 0.408·6-s + 0.737·7-s + 0.353·8-s + 0.333·9-s − 0.0614·10-s + 1.90·11-s − 0.288·12-s + 1.33·13-s + 0.521·14-s + 0.0501·15-s + 0.250·16-s − 0.778·17-s + 0.235·18-s + 0.197·19-s − 0.0434·20-s − 0.425·21-s + 1.34·22-s − 0.204·24-s − 0.992·25-s + 0.943·26-s − 0.192·27-s + 0.368·28-s + 1.00·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3174 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3174 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3174\)    =    \(2 \cdot 3 \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(25.3445\)
Root analytic conductor: \(5.03433\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3174,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.186787665\)
\(L(\frac12)\) \(\approx\) \(3.186787665\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
23 \( 1 \)
good5 \( 1 + 0.194T + 5T^{2} \)
7 \( 1 - 1.95T + 7T^{2} \)
11 \( 1 - 6.30T + 11T^{2} \)
13 \( 1 - 4.81T + 13T^{2} \)
17 \( 1 + 3.20T + 17T^{2} \)
19 \( 1 - 0.859T + 19T^{2} \)
29 \( 1 - 5.41T + 29T^{2} \)
31 \( 1 + 1.01T + 31T^{2} \)
37 \( 1 - 6.89T + 37T^{2} \)
41 \( 1 + 0.972T + 41T^{2} \)
43 \( 1 + 5.14T + 43T^{2} \)
47 \( 1 + 7.41T + 47T^{2} \)
53 \( 1 - 2.00T + 53T^{2} \)
59 \( 1 + 8.60T + 59T^{2} \)
61 \( 1 - 12.1T + 61T^{2} \)
67 \( 1 - 7.09T + 67T^{2} \)
71 \( 1 - 1.10T + 71T^{2} \)
73 \( 1 + 13.1T + 73T^{2} \)
79 \( 1 - 1.12T + 79T^{2} \)
83 \( 1 - 4.00T + 83T^{2} \)
89 \( 1 + 9.55T + 89T^{2} \)
97 \( 1 + 1.12T + 97T^{2} \)
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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

−8.602299094078072709854755219479, −7.904315204975310582739806058501, −6.77796755541019867278441694020, −6.41322948265320094130032900509, −5.70450439554827416314188231628, −4.66968881442184438520532032328, −4.12777972212021034289981633053, −3.38319761757124889788430098909, −1.88374512670597499121638051623, −1.11773203611966143124392243727, 1.11773203611966143124392243727, 1.88374512670597499121638051623, 3.38319761757124889788430098909, 4.12777972212021034289981633053, 4.66968881442184438520532032328, 5.70450439554827416314188231628, 6.41322948265320094130032900509, 6.77796755541019867278441694020, 7.904315204975310582739806058501, 8.602299094078072709854755219479

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