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Properties

Label 2-32340-1.1-c1-0-3
Degree $2$
Conductor $32340$
Sign $1$
Analytic cond. $258.236$
Root an. cond. $16.0697$
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  − 3-s + 5-s + 9-s − 11-s − 6·13-s − 15-s + 3·17-s + 19-s − 23-s + 25-s − 27-s + 5·29-s + 33-s − 7·37-s + 6·39-s − 10·41-s − 13·43-s + 45-s − 7·47-s − 3·51-s − 10·53-s − 55-s − 57-s + 13·59-s − 8·61-s − 6·65-s − 14·67-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 1/3·9-s − 0.301·11-s − 1.66·13-s − 0.258·15-s + 0.727·17-s + 0.229·19-s − 0.208·23-s + 1/5·25-s − 0.192·27-s + 0.928·29-s + 0.174·33-s − 1.15·37-s + 0.960·39-s − 1.56·41-s − 1.98·43-s + 0.149·45-s − 1.02·47-s − 0.420·51-s − 1.37·53-s − 0.134·55-s − 0.132·57-s + 1.69·59-s − 1.02·61-s − 0.744·65-s − 1.71·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 32340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32340 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(32340\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7^{2} \cdot 11\)
Sign: $1$
Analytic conductor: \(258.236\)
Root analytic conductor: \(16.0697\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 32340,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.016331219\)
\(L(\frac12)\) \(\approx\) \(1.016331219\)
\(L(\frac{3}{2})\) 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 \( 1 \)
11 \( 1 + T \)
good13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 7 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 13 T + p T^{2} \)
47 \( 1 + 7 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 13 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 14 T + p T^{2} \)
71 \( 1 - 5 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 16 T + p T^{2} \)
97 \( 1 - 17 T + p T^{2} \)
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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

−14.92164837281063, −14.64904509452399, −13.91042604951746, −13.52876857614725, −12.82765169329613, −12.34792075890582, −11.84207292113754, −11.56472374028292, −10.55536765363980, −10.21563441902234, −9.860984656446082, −9.295069623067853, −8.461477077933068, −7.955983441491068, −7.294032235674560, −6.748646326982940, −6.269254234133147, −5.422127176068683, −4.945070771462717, −4.722829527323680, −3.504298080429215, −3.044085477063828, −2.101073708096124, −1.526331643924162, −0.3866778341483701, 0.3866778341483701, 1.526331643924162, 2.101073708096124, 3.044085477063828, 3.504298080429215, 4.722829527323680, 4.945070771462717, 5.422127176068683, 6.269254234133147, 6.748646326982940, 7.294032235674560, 7.955983441491068, 8.461477077933068, 9.295069623067853, 9.860984656446082, 10.21563441902234, 10.55536765363980, 11.56472374028292, 11.84207292113754, 12.34792075890582, 12.82765169329613, 13.52876857614725, 13.91042604951746, 14.64904509452399, 14.92164837281063

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