Properties

Label 2-3174-1.1-c1-0-81
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 4-s − 1.23·5-s + 6-s − 1.47·7-s + 8-s + 9-s − 1.23·10-s − 0.886·11-s + 12-s + 0.0673·13-s − 1.47·14-s − 1.23·15-s + 16-s − 4.29·17-s + 18-s − 4.66·19-s − 1.23·20-s − 1.47·21-s − 0.886·22-s + 24-s − 3.47·25-s + 0.0673·26-s + 27-s − 1.47·28-s − 7.67·29-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.552·5-s + 0.408·6-s − 0.558·7-s + 0.353·8-s + 0.333·9-s − 0.391·10-s − 0.267·11-s + 0.288·12-s + 0.0186·13-s − 0.395·14-s − 0.319·15-s + 0.250·16-s − 1.04·17-s + 0.235·18-s − 1.07·19-s − 0.276·20-s − 0.322·21-s − 0.189·22-s + 0.204·24-s − 0.694·25-s + 0.0132·26-s + 0.192·27-s − 0.279·28-s − 1.42·29-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: \(1\)
Selberg data: \((2,\ 3174,\ (\ :1/2),\ -1)\)

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)$
bad2 \( 1 - T \)
3 \( 1 - T \)
23 \( 1 \)
good5 \( 1 + 1.23T + 5T^{2} \)
7 \( 1 + 1.47T + 7T^{2} \)
11 \( 1 + 0.886T + 11T^{2} \)
13 \( 1 - 0.0673T + 13T^{2} \)
17 \( 1 + 4.29T + 17T^{2} \)
19 \( 1 + 4.66T + 19T^{2} \)
29 \( 1 + 7.67T + 29T^{2} \)
31 \( 1 + 1.59T + 31T^{2} \)
37 \( 1 + 5.22T + 37T^{2} \)
41 \( 1 + 0.135T + 41T^{2} \)
43 \( 1 - 6.57T + 43T^{2} \)
47 \( 1 - 8.31T + 47T^{2} \)
53 \( 1 + 2.99T + 53T^{2} \)
59 \( 1 + 6.83T + 59T^{2} \)
61 \( 1 + 4.60T + 61T^{2} \)
67 \( 1 - 12.4T + 67T^{2} \)
71 \( 1 + 7.31T + 71T^{2} \)
73 \( 1 - 13.1T + 73T^{2} \)
79 \( 1 - 7.89T + 79T^{2} \)
83 \( 1 + 8.40T + 83T^{2} \)
89 \( 1 - 11.7T + 89T^{2} \)
97 \( 1 + 17.1T + 97T^{2} \)
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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

−8.185792349259756343660848567523, −7.51899654042896056955742177038, −6.77966204688557145370952880347, −6.07394853732968361638248770172, −5.12590415859182109837502982642, −4.12211893154684407536961724403, −3.73357558457842546100499645981, −2.67901505289397348183946935923, −1.88004260275575038833982712305, 0, 1.88004260275575038833982712305, 2.67901505289397348183946935923, 3.73357558457842546100499645981, 4.12211893154684407536961724403, 5.12590415859182109837502982642, 6.07394853732968361638248770172, 6.77966204688557145370952880347, 7.51899654042896056955742177038, 8.185792349259756343660848567523

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