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Properties

Label 2-80850-1.1-c1-0-108
Degree $2$
Conductor $80850$
Sign $-1$
Analytic cond. $645.590$
Root an. cond. $25.4084$
Motivic weight $1$
Arithmetic yes
Rational yes
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 − 6-s − 8-s + 9-s − 11-s + 12-s − 6·13-s + 16-s − 6·17-s − 18-s + 4·19-s + 22-s + 4·23-s − 24-s + 6·26-s + 27-s + 2·29-s − 32-s − 33-s + 6·34-s + 36-s − 2·37-s − 4·38-s − 6·39-s + 2·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.301·11-s + 0.288·12-s − 1.66·13-s + 1/4·16-s − 1.45·17-s − 0.235·18-s + 0.917·19-s + 0.213·22-s + 0.834·23-s − 0.204·24-s + 1.17·26-s + 0.192·27-s + 0.371·29-s − 0.176·32-s − 0.174·33-s + 1.02·34-s + 1/6·36-s − 0.328·37-s − 0.648·38-s − 0.960·39-s + 0.312·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80850\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7^{2} \cdot 11\)
Sign: $-1$
Analytic conductor: \(645.590\)
Root analytic conductor: \(25.4084\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 80850,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
5 \( 1 \)
7 \( 1 \)
11 \( 1 + T \)
good13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−14.29914015314029, −13.81087174940004, −13.15042309987506, −12.82540401921857, −12.11352289369306, −11.84584063804988, −11.00654988356650, −10.79818064980265, −10.08240532590672, −9.523263166946380, −9.361408206780718, −8.696234771594398, −8.220514631015646, −7.531369349137483, −7.304755085971842, −6.733140108114601, −6.164848816932499, −5.307391456026555, −4.786931695614298, −4.371658653491673, −3.337982177669865, −2.912651922073053, −2.286942954987652, −1.790510691545413, −0.8125703062574028, 0, 0.8125703062574028, 1.790510691545413, 2.286942954987652, 2.912651922073053, 3.337982177669865, 4.371658653491673, 4.786931695614298, 5.307391456026555, 6.164848816932499, 6.733140108114601, 7.304755085971842, 7.531369349137483, 8.220514631015646, 8.696234771594398, 9.361408206780718, 9.523263166946380, 10.08240532590672, 10.79818064980265, 11.00654988356650, 11.84584063804988, 12.11352289369306, 12.82540401921857, 13.15042309987506, 13.81087174940004, 14.29914015314029

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