Properties

Label 2-8035-1.1-c1-0-224
Degree $2$
Conductor $8035$
Sign $1$
Analytic cond. $64.1597$
Root an. cond. $8.00998$
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  − 2.41·2-s + 3.20·3-s + 3.85·4-s + 5-s − 7.76·6-s − 2.07·7-s − 4.47·8-s + 7.29·9-s − 2.41·10-s − 4.56·11-s + 12.3·12-s + 3.08·13-s + 5.02·14-s + 3.20·15-s + 3.12·16-s + 3.05·17-s − 17.6·18-s − 0.524·19-s + 3.85·20-s − 6.67·21-s + 11.0·22-s + 7.45·23-s − 14.3·24-s + 25-s − 7.45·26-s + 13.7·27-s − 8.00·28-s + ⋯
L(s)  = 1  − 1.71·2-s + 1.85·3-s + 1.92·4-s + 0.447·5-s − 3.16·6-s − 0.785·7-s − 1.58·8-s + 2.43·9-s − 0.764·10-s − 1.37·11-s + 3.56·12-s + 0.854·13-s + 1.34·14-s + 0.828·15-s + 0.780·16-s + 0.740·17-s − 4.15·18-s − 0.120·19-s + 0.860·20-s − 1.45·21-s + 2.35·22-s + 1.55·23-s − 2.93·24-s + 0.200·25-s − 1.46·26-s + 2.65·27-s − 1.51·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8035\)    =    \(5 \cdot 1607\)
Sign: $1$
Analytic conductor: \(64.1597\)
Root analytic conductor: \(8.00998\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8035,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.980565428\)
\(L(\frac12)\) \(\approx\) \(1.980565428\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - T \)
1607 \( 1 + T \)
good2 \( 1 + 2.41T + 2T^{2} \)
3 \( 1 - 3.20T + 3T^{2} \)
7 \( 1 + 2.07T + 7T^{2} \)
11 \( 1 + 4.56T + 11T^{2} \)
13 \( 1 - 3.08T + 13T^{2} \)
17 \( 1 - 3.05T + 17T^{2} \)
19 \( 1 + 0.524T + 19T^{2} \)
23 \( 1 - 7.45T + 23T^{2} \)
29 \( 1 - 2.84T + 29T^{2} \)
31 \( 1 + 6.26T + 31T^{2} \)
37 \( 1 - 5.05T + 37T^{2} \)
41 \( 1 - 1.86T + 41T^{2} \)
43 \( 1 - 7.79T + 43T^{2} \)
47 \( 1 - 2.35T + 47T^{2} \)
53 \( 1 + 7.65T + 53T^{2} \)
59 \( 1 + 1.36T + 59T^{2} \)
61 \( 1 - 4.00T + 61T^{2} \)
67 \( 1 - 3.62T + 67T^{2} \)
71 \( 1 + 9.13T + 71T^{2} \)
73 \( 1 - 16.3T + 73T^{2} \)
79 \( 1 - 2.32T + 79T^{2} \)
83 \( 1 + 6.26T + 83T^{2} \)
89 \( 1 - 6.63T + 89T^{2} \)
97 \( 1 + 6.71T + 97T^{2} \)
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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

−7.970429415048605187369178049390, −7.55209163652292691448603767462, −6.90973333883620269486291008787, −6.15391289957969471026508624698, −5.07101507635262849761163175231, −3.84782933226584520208236313979, −2.92976929004825729988890971485, −2.67508040948752157209595483753, −1.72308374017096966095526991307, −0.842938425027911934586699327242, 0.842938425027911934586699327242, 1.72308374017096966095526991307, 2.67508040948752157209595483753, 2.92976929004825729988890971485, 3.84782933226584520208236313979, 5.07101507635262849761163175231, 6.15391289957969471026508624698, 6.90973333883620269486291008787, 7.55209163652292691448603767462, 7.970429415048605187369178049390

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