Properties

Label 2-3174-1.1-c1-0-17
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 − 3.88·5-s − 6-s + 4.16·7-s + 8-s + 9-s − 3.88·10-s + 4.52·11-s − 12-s − 3.26·13-s + 4.16·14-s + 3.88·15-s + 16-s − 1.82·17-s + 18-s + 1.36·19-s − 3.88·20-s − 4.16·21-s + 4.52·22-s − 24-s + 10.1·25-s − 3.26·26-s − 27-s + 4.16·28-s + 0.561·29-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.73·5-s − 0.408·6-s + 1.57·7-s + 0.353·8-s + 0.333·9-s − 1.22·10-s + 1.36·11-s − 0.288·12-s − 0.905·13-s + 1.11·14-s + 1.00·15-s + 0.250·16-s − 0.443·17-s + 0.235·18-s + 0.313·19-s − 0.868·20-s − 0.908·21-s + 0.965·22-s − 0.204·24-s + 2.02·25-s − 0.639·26-s − 0.192·27-s + 0.786·28-s + 0.104·29-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

\(L(1)\) \(\approx\) \(2.144451090\)
\(L(\frac12)\) \(\approx\) \(2.144451090\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.88T + 5T^{2} \)
7 \( 1 - 4.16T + 7T^{2} \)
11 \( 1 - 4.52T + 11T^{2} \)
13 \( 1 + 3.26T + 13T^{2} \)
17 \( 1 + 1.82T + 17T^{2} \)
19 \( 1 - 1.36T + 19T^{2} \)
29 \( 1 - 0.561T + 29T^{2} \)
31 \( 1 + 4.63T + 31T^{2} \)
37 \( 1 - 2.51T + 37T^{2} \)
41 \( 1 + 6.11T + 41T^{2} \)
43 \( 1 - 9.85T + 43T^{2} \)
47 \( 1 - 2.38T + 47T^{2} \)
53 \( 1 - 10.6T + 53T^{2} \)
59 \( 1 + 7.15T + 59T^{2} \)
61 \( 1 + 3.25T + 61T^{2} \)
67 \( 1 + 6.28T + 67T^{2} \)
71 \( 1 + 6.45T + 71T^{2} \)
73 \( 1 - 12.3T + 73T^{2} \)
79 \( 1 - 8.43T + 79T^{2} \)
83 \( 1 + 0.377T + 83T^{2} \)
89 \( 1 - 15.2T + 89T^{2} \)
97 \( 1 - 18.8T + 97T^{2} \)
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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

−8.516173054932348241854629552327, −7.53968737914825810258670303012, −7.40000041058828871770738285159, −6.45154703059529614958060481241, −5.37333940667583489824377953342, −4.59281041218632580755019914458, −4.26074983321649381191962452567, −3.42917223512504785106090752471, −2.02671707506926982232856938921, −0.845761601078044505215514377422, 0.845761601078044505215514377422, 2.02671707506926982232856938921, 3.42917223512504785106090752471, 4.26074983321649381191962452567, 4.59281041218632580755019914458, 5.37333940667583489824377953342, 6.45154703059529614958060481241, 7.40000041058828871770738285159, 7.53968737914825810258670303012, 8.516173054932348241854629552327

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