Properties

Label 2-3174-1.1-c1-0-46
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 + 3.82·5-s − 6-s + 2.89·7-s + 8-s + 9-s + 3.82·10-s − 1.58·11-s − 12-s + 4.70·13-s + 2.89·14-s − 3.82·15-s + 16-s − 4.73·17-s + 18-s + 2.14·19-s + 3.82·20-s − 2.89·21-s − 1.58·22-s − 24-s + 9.61·25-s + 4.70·26-s − 27-s + 2.89·28-s − 0.588·29-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.70·5-s − 0.408·6-s + 1.09·7-s + 0.353·8-s + 0.333·9-s + 1.20·10-s − 0.476·11-s − 0.288·12-s + 1.30·13-s + 0.773·14-s − 0.987·15-s + 0.250·16-s − 1.14·17-s + 0.235·18-s + 0.492·19-s + 0.854·20-s − 0.631·21-s − 0.336·22-s − 0.204·24-s + 1.92·25-s + 0.923·26-s − 0.192·27-s + 0.546·28-s − 0.109·29-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

\(L(1)\) \(\approx\) \(4.039035998\)
\(L(\frac12)\) \(\approx\) \(4.039035998\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.82T + 5T^{2} \)
7 \( 1 - 2.89T + 7T^{2} \)
11 \( 1 + 1.58T + 11T^{2} \)
13 \( 1 - 4.70T + 13T^{2} \)
17 \( 1 + 4.73T + 17T^{2} \)
19 \( 1 - 2.14T + 19T^{2} \)
29 \( 1 + 0.588T + 29T^{2} \)
31 \( 1 - 0.170T + 31T^{2} \)
37 \( 1 - 4.81T + 37T^{2} \)
41 \( 1 + 10.3T + 41T^{2} \)
43 \( 1 - 12.2T + 43T^{2} \)
47 \( 1 - 5.45T + 47T^{2} \)
53 \( 1 + 10.3T + 53T^{2} \)
59 \( 1 + 1.90T + 59T^{2} \)
61 \( 1 + 12.6T + 61T^{2} \)
67 \( 1 - 5.87T + 67T^{2} \)
71 \( 1 + 11.8T + 71T^{2} \)
73 \( 1 + 13.2T + 73T^{2} \)
79 \( 1 - 11.6T + 79T^{2} \)
83 \( 1 + 0.113T + 83T^{2} \)
89 \( 1 + 12.2T + 89T^{2} \)
97 \( 1 - 0.873T + 97T^{2} \)
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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

−8.750059625141416077314355439888, −7.83353644918867098351143738889, −6.87474490998796393406251352750, −6.08380442760485243754766408569, −5.72503650876245686116739784682, −4.92783476074500827060415491045, −4.30634250091168137008707925712, −2.95189047728993086517419247383, −1.95957502555241921865406795984, −1.29846118976158710537875400389, 1.29846118976158710537875400389, 1.95957502555241921865406795984, 2.95189047728993086517419247383, 4.30634250091168137008707925712, 4.92783476074500827060415491045, 5.72503650876245686116739784682, 6.08380442760485243754766408569, 6.87474490998796393406251352750, 7.83353644918867098351143738889, 8.750059625141416077314355439888

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