Properties

Label 2-229320-1.1-c1-0-75
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 − 3·11-s − 13-s + 3·17-s + 23-s + 25-s − 6·29-s + 10·31-s − 9·37-s + 3·41-s − 10·43-s + 2·47-s − 10·53-s + 3·55-s − 9·59-s + 2·61-s + 65-s − 5·67-s + 7·73-s − 79-s + 10·83-s − 3·85-s − 3·89-s − 7·97-s + 101-s + 103-s + 107-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.904·11-s − 0.277·13-s + 0.727·17-s + 0.208·23-s + 1/5·25-s − 1.11·29-s + 1.79·31-s − 1.47·37-s + 0.468·41-s − 1.52·43-s + 0.291·47-s − 1.37·53-s + 0.404·55-s − 1.17·59-s + 0.256·61-s + 0.124·65-s − 0.610·67-s + 0.819·73-s − 0.112·79-s + 1.09·83-s − 0.325·85-s − 0.317·89-s − 0.710·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-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: \(1\)
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 \( 1 + 3 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 + 9 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 9 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 5 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 7 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 - 10 T + p T^{2} \)
89 \( 1 + 3 T + p T^{2} \)
97 \( 1 + 7 T + p T^{2} \)
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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

−13.19290708166357, −12.58625501837335, −12.25356750509529, −11.85647521246571, −11.31506460148921, −10.77875962312959, −10.46574507437951, −9.864273276796783, −9.551280755861637, −8.912620841622720, −8.312153178218013, −8.004009907039965, −7.574382788241299, −7.019634144735394, −6.566942203054436, −5.906505462604163, −5.439937877043795, −4.848119188828490, −4.577366746138558, −3.768786088139613, −3.211169180564492, −2.900732060319479, −2.080218027698897, −1.528505659668996, −0.6772048347992868, 0, 0.6772048347992868, 1.528505659668996, 2.080218027698897, 2.900732060319479, 3.211169180564492, 3.768786088139613, 4.577366746138558, 4.848119188828490, 5.439937877043795, 5.906505462604163, 6.566942203054436, 7.019634144735394, 7.574382788241299, 8.004009907039965, 8.312153178218013, 8.912620841622720, 9.551280755861637, 9.864273276796783, 10.46574507437951, 10.77875962312959, 11.31506460148921, 11.85647521246571, 12.25356750509529, 12.58625501837335, 13.19290708166357

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