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Properties

Label 2-32368-1.1-c1-0-6
Degree $2$
Conductor $32368$
Sign $1$
Analytic cond. $258.459$
Root an. cond. $16.0766$
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  − 7-s − 3·9-s + 2·13-s + 4·19-s + 8·23-s − 5·25-s − 8·29-s + 8·31-s + 8·37-s − 8·41-s − 4·43-s + 8·47-s + 49-s + 10·53-s − 12·59-s + 3·63-s + 4·67-s − 8·71-s − 8·73-s + 8·79-s + 9·81-s − 4·83-s + 6·89-s − 2·91-s − 8·97-s + 101-s + 103-s + ⋯
L(s)  = 1  − 0.377·7-s − 9-s + 0.554·13-s + 0.917·19-s + 1.66·23-s − 25-s − 1.48·29-s + 1.43·31-s + 1.31·37-s − 1.24·41-s − 0.609·43-s + 1.16·47-s + 1/7·49-s + 1.37·53-s − 1.56·59-s + 0.377·63-s + 0.488·67-s − 0.949·71-s − 0.936·73-s + 0.900·79-s + 81-s − 0.439·83-s + 0.635·89-s − 0.209·91-s − 0.812·97-s + 0.0995·101-s + 0.0985·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 32368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(32368\)    =    \(2^{4} \cdot 7 \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(258.459\)
Root analytic conductor: \(16.0766\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 32368,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.825835903\)
\(L(\frac12)\) \(\approx\) \(1.825835903\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + T \)
17 \( 1 \)
good3 \( 1 + p T^{2} \)
5 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−15.20696157225448, −14.56774909459357, −13.82576802053975, −13.47693484404024, −13.15391709575768, −12.32545000731696, −11.71853253108962, −11.45117832660870, −10.85473012854773, −10.22695046398621, −9.612702801171319, −9.061889590487520, −8.680067746872005, −7.923877357434579, −7.456402358013816, −6.767931842866689, −6.103385190573399, −5.648618507354211, −5.078394454010503, −4.294006840084432, −3.499252982015980, −3.043243550478541, −2.370528441487840, −1.360663149707892, −0.5360054065281212, 0.5360054065281212, 1.360663149707892, 2.370528441487840, 3.043243550478541, 3.499252982015980, 4.294006840084432, 5.078394454010503, 5.648618507354211, 6.103385190573399, 6.767931842866689, 7.456402358013816, 7.923877357434579, 8.680067746872005, 9.061889590487520, 9.612702801171319, 10.22695046398621, 10.85473012854773, 11.45117832660870, 11.71853253108962, 12.32545000731696, 13.15391709575768, 13.47693484404024, 13.82576802053975, 14.56774909459357, 15.20696157225448

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