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Properties

Label 2-203280-1.1-c1-0-175
Degree $2$
Conductor $203280$
Sign $-1$
Analytic cond. $1623.19$
Root an. cond. $40.2889$
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  − 3-s + 5-s + 7-s + 9-s − 2·13-s − 15-s + 6·17-s + 8·19-s − 21-s + 25-s − 27-s − 6·29-s + 4·31-s + 35-s − 10·37-s + 2·39-s + 6·41-s − 4·43-s + 45-s + 49-s − 6·51-s − 6·53-s − 8·57-s + 12·59-s + 10·61-s + 63-s − 2·65-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s − 0.554·13-s − 0.258·15-s + 1.45·17-s + 1.83·19-s − 0.218·21-s + 1/5·25-s − 0.192·27-s − 1.11·29-s + 0.718·31-s + 0.169·35-s − 1.64·37-s + 0.320·39-s + 0.937·41-s − 0.609·43-s + 0.149·45-s + 1/7·49-s − 0.840·51-s − 0.824·53-s − 1.05·57-s + 1.56·59-s + 1.28·61-s + 0.125·63-s − 0.248·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 203280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 203280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(203280\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7 \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(1623.19\)
Root analytic conductor: \(40.2889\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 203280,\ (\ :1/2),\ -1)\)

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 \( 1 \)
3 \( 1 + T \)
5 \( 1 - T \)
7 \( 1 - T \)
11 \( 1 \)
good13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−13.35656147645669, −12.64248091715782, −12.30365029564277, −11.89746100815711, −11.45468807833298, −11.03050037589542, −10.38709798296729, −9.955437395449198, −9.649805849733649, −9.226245550711773, −8.485410374822239, −7.945672278095517, −7.531085464096428, −7.059504042331640, −6.612413562330844, −5.800530886993757, −5.497775693347449, −5.123481712589647, −4.706475275483885, −3.676650235058278, −3.542383260124910, −2.692139708250000, −2.107636884117571, −1.282039160481259, −0.9976935645698564, 0, 0.9976935645698564, 1.282039160481259, 2.107636884117571, 2.692139708250000, 3.542383260124910, 3.676650235058278, 4.706475275483885, 5.123481712589647, 5.497775693347449, 5.800530886993757, 6.612413562330844, 7.059504042331640, 7.531085464096428, 7.945672278095517, 8.485410374822239, 9.226245550711773, 9.649805849733649, 9.955437395449198, 10.38709798296729, 11.03050037589542, 11.45468807833298, 11.89746100815711, 12.30365029564277, 12.64248091715782, 13.35656147645669

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