Properties

Label 2-354-1.1-c7-0-42
Degree $2$
Conductor $354$
Sign $-1$
Analytic cond. $110.584$
Root an. cond. $10.5159$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 8·2-s − 27·3-s + 64·4-s − 490.·5-s − 216·6-s + 975.·7-s + 512·8-s + 729·9-s − 3.92e3·10-s − 7.23e3·11-s − 1.72e3·12-s + 5.31e3·13-s + 7.80e3·14-s + 1.32e4·15-s + 4.09e3·16-s + 6.97e3·17-s + 5.83e3·18-s + 4.12e4·19-s − 3.14e4·20-s − 2.63e4·21-s − 5.79e4·22-s − 3.38e4·23-s − 1.38e4·24-s + 1.62e5·25-s + 4.25e4·26-s − 1.96e4·27-s + 6.24e4·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.75·5-s − 0.408·6-s + 1.07·7-s + 0.353·8-s + 0.333·9-s − 1.24·10-s − 1.63·11-s − 0.288·12-s + 0.671·13-s + 0.759·14-s + 1.01·15-s + 0.250·16-s + 0.344·17-s + 0.235·18-s + 1.37·19-s − 0.877·20-s − 0.620·21-s − 1.15·22-s − 0.580·23-s − 0.204·24-s + 2.08·25-s + 0.474·26-s − 0.192·27-s + 0.537·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(354\)    =    \(2 \cdot 3 \cdot 59\)
Sign: $-1$
Analytic conductor: \(110.584\)
Root analytic conductor: \(10.5159\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 354,\ (\ :7/2),\ -1)\)

Particular Values

\(L(4)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 8T \)
3 \( 1 + 27T \)
59 \( 1 + 2.05e5T \)
good5 \( 1 + 490.T + 7.81e4T^{2} \)
7 \( 1 - 975.T + 8.23e5T^{2} \)
11 \( 1 + 7.23e3T + 1.94e7T^{2} \)
13 \( 1 - 5.31e3T + 6.27e7T^{2} \)
17 \( 1 - 6.97e3T + 4.10e8T^{2} \)
19 \( 1 - 4.12e4T + 8.93e8T^{2} \)
23 \( 1 + 3.38e4T + 3.40e9T^{2} \)
29 \( 1 - 6.54e4T + 1.72e10T^{2} \)
31 \( 1 - 5.89e4T + 2.75e10T^{2} \)
37 \( 1 + 2.71e5T + 9.49e10T^{2} \)
41 \( 1 + 5.31e5T + 1.94e11T^{2} \)
43 \( 1 - 2.69e5T + 2.71e11T^{2} \)
47 \( 1 - 8.40e5T + 5.06e11T^{2} \)
53 \( 1 - 4.30e5T + 1.17e12T^{2} \)
61 \( 1 + 2.69e5T + 3.14e12T^{2} \)
67 \( 1 - 1.59e6T + 6.06e12T^{2} \)
71 \( 1 + 2.78e6T + 9.09e12T^{2} \)
73 \( 1 + 1.90e6T + 1.10e13T^{2} \)
79 \( 1 + 4.73e6T + 1.92e13T^{2} \)
83 \( 1 + 8.24e6T + 2.71e13T^{2} \)
89 \( 1 - 2.74e6T + 4.42e13T^{2} \)
97 \( 1 - 7.05e6T + 8.07e13T^{2} \)
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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

−10.32586433767314265366549219450, −8.412305402286243125043950988345, −7.80344205849290973206664605270, −7.12625085562337011309771268168, −5.56249279832283564346921587509, −4.88653917483137975560896353420, −3.96265535939508839229899682583, −2.90220618857152639681191828578, −1.21569883211088824723722201094, 0, 1.21569883211088824723722201094, 2.90220618857152639681191828578, 3.96265535939508839229899682583, 4.88653917483137975560896353420, 5.56249279832283564346921587509, 7.12625085562337011309771268168, 7.80344205849290973206664605270, 8.412305402286243125043950988345, 10.32586433767314265366549219450

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