Properties

Label 2-6045-1.1-c1-0-14
Degree $2$
Conductor $6045$
Sign $1$
Analytic cond. $48.2695$
Root an. cond. $6.94763$
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  − 2.74·2-s − 3-s + 5.55·4-s − 5-s + 2.74·6-s − 0.783·7-s − 9.77·8-s + 9-s + 2.74·10-s − 4.40·11-s − 5.55·12-s − 13-s + 2.15·14-s + 15-s + 15.7·16-s − 4.56·17-s − 2.74·18-s + 7.83·19-s − 5.55·20-s + 0.783·21-s + 12.1·22-s + 2.11·23-s + 9.77·24-s + 25-s + 2.74·26-s − 27-s − 4.35·28-s + ⋯
L(s)  = 1  − 1.94·2-s − 0.577·3-s + 2.77·4-s − 0.447·5-s + 1.12·6-s − 0.296·7-s − 3.45·8-s + 0.333·9-s + 0.869·10-s − 1.32·11-s − 1.60·12-s − 0.277·13-s + 0.575·14-s + 0.258·15-s + 3.94·16-s − 1.10·17-s − 0.647·18-s + 1.79·19-s − 1.24·20-s + 0.170·21-s + 2.58·22-s + 0.442·23-s + 1.99·24-s + 0.200·25-s + 0.539·26-s − 0.192·27-s − 0.822·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6045\)    =    \(3 \cdot 5 \cdot 13 \cdot 31\)
Sign: $1$
Analytic conductor: \(48.2695\)
Root analytic conductor: \(6.94763\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6045,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2751144182\)
\(L(\frac12)\) \(\approx\) \(0.2751144182\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
5 \( 1 + T \)
13 \( 1 + T \)
31 \( 1 - T \)
good2 \( 1 + 2.74T + 2T^{2} \)
7 \( 1 + 0.783T + 7T^{2} \)
11 \( 1 + 4.40T + 11T^{2} \)
17 \( 1 + 4.56T + 17T^{2} \)
19 \( 1 - 7.83T + 19T^{2} \)
23 \( 1 - 2.11T + 23T^{2} \)
29 \( 1 - 0.715T + 29T^{2} \)
37 \( 1 + 2.53T + 37T^{2} \)
41 \( 1 - 7.47T + 41T^{2} \)
43 \( 1 - 8.74T + 43T^{2} \)
47 \( 1 + 0.973T + 47T^{2} \)
53 \( 1 - 4.11T + 53T^{2} \)
59 \( 1 + 2.81T + 59T^{2} \)
61 \( 1 + 10.4T + 61T^{2} \)
67 \( 1 + 11.6T + 67T^{2} \)
71 \( 1 + 2.44T + 71T^{2} \)
73 \( 1 + 5.12T + 73T^{2} \)
79 \( 1 + 16.3T + 79T^{2} \)
83 \( 1 + 14.8T + 83T^{2} \)
89 \( 1 - 2.18T + 89T^{2} \)
97 \( 1 + 8.60T + 97T^{2} \)
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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

−8.013783623368505776450800910614, −7.38051060309828261745167673261, −7.17244890181556580745429370357, −6.15458039009633087286742684883, −5.55446098993050394818726983024, −4.54943494593234181435243475672, −3.10768693220123998386475797708, −2.60483658521979280295764257499, −1.41449486440582571734141478384, −0.39482895639102895654281415338, 0.39482895639102895654281415338, 1.41449486440582571734141478384, 2.60483658521979280295764257499, 3.10768693220123998386475797708, 4.54943494593234181435243475672, 5.55446098993050394818726983024, 6.15458039009633087286742684883, 7.17244890181556580745429370357, 7.38051060309828261745167673261, 8.013783623368505776450800910614

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