Properties

Label 2-229320-1.1-c1-0-89
Degree $2$
Conductor $229320$
Sign $-1$
Analytic cond. $1831.12$
Root an. cond. $42.7916$
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  + 5-s − 13-s − 2·17-s − 2·19-s − 6·23-s + 25-s + 2·29-s − 6·31-s + 6·37-s − 10·41-s − 4·43-s + 12·53-s − 12·59-s − 65-s + 16·67-s − 8·71-s + 14·73-s − 4·83-s − 2·85-s + 6·89-s − 2·95-s + 10·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.277·13-s − 0.485·17-s − 0.458·19-s − 1.25·23-s + 1/5·25-s + 0.371·29-s − 1.07·31-s + 0.986·37-s − 1.56·41-s − 0.609·43-s + 1.64·53-s − 1.56·59-s − 0.124·65-s + 1.95·67-s − 0.949·71-s + 1.63·73-s − 0.439·83-s − 0.216·85-s + 0.635·89-s − 0.205·95-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(229320\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 7^{2} \cdot 13\)
Sign: $-1$
Analytic conductor: \(1831.12\)
Root analytic conductor: \(42.7916\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 229320,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 - T \)
7 \( 1 \)
13 \( 1 + T \)
good11 \( 1 + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 16 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
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

−13.12673290952895, −12.82157332118829, −12.14233347320047, −11.87370377117376, −11.32649489939382, −10.72265862578383, −10.45114963394925, −9.767181579948391, −9.620505593559337, −8.876710512664346, −8.541427143106358, −7.985729878080452, −7.530874741067508, −6.870919472509961, −6.499313816031756, −6.032011277862748, −5.416746456555699, −5.024301017846202, −4.370865412567449, −3.894319582943913, −3.325837860172348, −2.609887967384807, −2.052944947027862, −1.679623310474213, −0.7330823171911563, 0, 0.7330823171911563, 1.679623310474213, 2.052944947027862, 2.609887967384807, 3.325837860172348, 3.894319582943913, 4.370865412567449, 5.024301017846202, 5.416746456555699, 6.032011277862748, 6.499313816031756, 6.870919472509961, 7.530874741067508, 7.985729878080452, 8.541427143106358, 8.876710512664346, 9.620505593559337, 9.767181579948391, 10.45114963394925, 10.72265862578383, 11.32649489939382, 11.87370377117376, 12.14233347320047, 12.82157332118829, 13.12673290952895

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