Properties

Label 2-3174-1.1-c1-0-34
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 − 1.06·5-s − 6-s + 4.42·7-s + 8-s + 9-s − 1.06·10-s − 1.64·11-s − 12-s + 5.47·13-s + 4.42·14-s + 1.06·15-s + 16-s + 3.92·17-s + 18-s + 6.60·19-s − 1.06·20-s − 4.42·21-s − 1.64·22-s − 24-s − 3.86·25-s + 5.47·26-s − 27-s + 4.42·28-s − 6.39·29-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.477·5-s − 0.408·6-s + 1.67·7-s + 0.353·8-s + 0.333·9-s − 0.337·10-s − 0.497·11-s − 0.288·12-s + 1.51·13-s + 1.18·14-s + 0.275·15-s + 0.250·16-s + 0.952·17-s + 0.235·18-s + 1.51·19-s − 0.238·20-s − 0.965·21-s − 0.351·22-s − 0.204·24-s − 0.772·25-s + 1.07·26-s − 0.192·27-s + 0.836·28-s − 1.18·29-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

\(L(1)\) \(\approx\) \(3.048076457\)
\(L(\frac12)\) \(\approx\) \(3.048076457\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.06T + 5T^{2} \)
7 \( 1 - 4.42T + 7T^{2} \)
11 \( 1 + 1.64T + 11T^{2} \)
13 \( 1 - 5.47T + 13T^{2} \)
17 \( 1 - 3.92T + 17T^{2} \)
19 \( 1 - 6.60T + 19T^{2} \)
29 \( 1 + 6.39T + 29T^{2} \)
31 \( 1 + 3.01T + 31T^{2} \)
37 \( 1 + 2.21T + 37T^{2} \)
41 \( 1 - 4.09T + 41T^{2} \)
43 \( 1 + 2.91T + 43T^{2} \)
47 \( 1 + 9.62T + 47T^{2} \)
53 \( 1 + 2.23T + 53T^{2} \)
59 \( 1 - 9.75T + 59T^{2} \)
61 \( 1 + 1.55T + 61T^{2} \)
67 \( 1 - 12.0T + 67T^{2} \)
71 \( 1 + 5.26T + 71T^{2} \)
73 \( 1 - 7.18T + 73T^{2} \)
79 \( 1 - 2.69T + 79T^{2} \)
83 \( 1 - 5.53T + 83T^{2} \)
89 \( 1 - 11.3T + 89T^{2} \)
97 \( 1 - 11.9T + 97T^{2} \)
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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

−8.338514534744632209554320676167, −7.81680185955254045913030515997, −7.30897824449940025241506195808, −6.16570161714789820865899158860, −5.34787759628515287955077279236, −5.10090520539402618731185514641, −3.95952470936422336089916876988, −3.42012626546148612207520117061, −1.91071301204308366097790381528, −1.07773873977739331198708438005, 1.07773873977739331198708438005, 1.91071301204308366097790381528, 3.42012626546148612207520117061, 3.95952470936422336089916876988, 5.10090520539402618731185514641, 5.34787759628515287955077279236, 6.16570161714789820865899158860, 7.30897824449940025241506195808, 7.81680185955254045913030515997, 8.338514534744632209554320676167

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