Properties

Label 2-229320-1.1-c1-0-13
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 11-s − 13-s − 2·17-s − 2·19-s + 25-s − 4·29-s − 4·31-s + 7·37-s + 10·41-s + 4·43-s − 12·47-s − 5·53-s + 55-s + 15·59-s − 3·61-s + 65-s + 3·67-s + 9·71-s − 9·73-s + 79-s + 14·83-s + 2·85-s + 13·89-s + 2·95-s − 2·97-s + 101-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.301·11-s − 0.277·13-s − 0.485·17-s − 0.458·19-s + 1/5·25-s − 0.742·29-s − 0.718·31-s + 1.15·37-s + 1.56·41-s + 0.609·43-s − 1.75·47-s − 0.686·53-s + 0.134·55-s + 1.95·59-s − 0.384·61-s + 0.124·65-s + 0.366·67-s + 1.06·71-s − 1.05·73-s + 0.112·79-s + 1.53·83-s + 0.216·85-s + 1.37·89-s + 0.205·95-s − 0.203·97-s + 0.0995·101-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: $\chi_{229320} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 229320,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.296048919\)
\(L(\frac12)\) \(\approx\) \(1.296048919\)
\(L(\frac{3}{2})\) 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 \( 1 + T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 7 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 5 T + p T^{2} \)
59 \( 1 - 15 T + p T^{2} \)
61 \( 1 + 3 T + p T^{2} \)
67 \( 1 - 3 T + p T^{2} \)
71 \( 1 - 9 T + p T^{2} \)
73 \( 1 + 9 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
89 \( 1 - 13 T + p T^{2} \)
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

−12.92126541474521, −12.59429340026181, −11.98229460753500, −11.46419288257144, −11.05437135358719, −10.80015400240125, −10.12111167008081, −9.623102009836780, −9.183998930215890, −8.784137345394623, −8.020739470147156, −7.835263385563981, −7.342538402547580, −6.657439612264084, −6.356618378853668, −5.667671007121556, −5.191386713947417, −4.640867716698004, −4.089768743174056, −3.694862940604644, −2.978510850821683, −2.392901302826223, −1.932132680693197, −1.055416483144754, −0.3437829451411921, 0.3437829451411921, 1.055416483144754, 1.932132680693197, 2.392901302826223, 2.978510850821683, 3.694862940604644, 4.089768743174056, 4.640867716698004, 5.191386713947417, 5.667671007121556, 6.356618378853668, 6.657439612264084, 7.342538402547580, 7.835263385563981, 8.020739470147156, 8.784137345394623, 9.183998930215890, 9.623102009836780, 10.12111167008081, 10.80015400240125, 11.05437135358719, 11.46419288257144, 11.98229460753500, 12.59429340026181, 12.92126541474521

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