Properties

Label 2-294-1.1-c5-0-14
Degree $2$
Conductor $294$
Sign $1$
Analytic cond. $47.1528$
Root an. cond. $6.86679$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s − 9·3-s + 16·4-s + 61.0·5-s − 36·6-s + 64·8-s + 81·9-s + 244.·10-s − 36.5·11-s − 144·12-s − 34.5·13-s − 549.·15-s + 256·16-s + 2.06e3·17-s + 324·18-s − 452.·19-s + 977.·20-s − 146.·22-s + 1.68e3·23-s − 576·24-s + 604.·25-s − 138.·26-s − 729·27-s − 4.76e3·29-s − 2.19e3·30-s + 5.26e3·31-s + 1.02e3·32-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.09·5-s − 0.408·6-s + 0.353·8-s + 0.333·9-s + 0.772·10-s − 0.0910·11-s − 0.288·12-s − 0.0567·13-s − 0.630·15-s + 0.250·16-s + 1.72·17-s + 0.235·18-s − 0.287·19-s + 0.546·20-s − 0.0643·22-s + 0.663·23-s − 0.204·24-s + 0.193·25-s − 0.0401·26-s − 0.192·27-s − 1.05·29-s − 0.446·30-s + 0.983·31-s + 0.176·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(294\)    =    \(2 \cdot 3 \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(47.1528\)
Root analytic conductor: \(6.86679\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 294,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(3.570386736\)
\(L(\frac12)\) \(\approx\) \(3.570386736\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 4T \)
3 \( 1 + 9T \)
7 \( 1 \)
good5 \( 1 - 61.0T + 3.12e3T^{2} \)
11 \( 1 + 36.5T + 1.61e5T^{2} \)
13 \( 1 + 34.5T + 3.71e5T^{2} \)
17 \( 1 - 2.06e3T + 1.41e6T^{2} \)
19 \( 1 + 452.T + 2.47e6T^{2} \)
23 \( 1 - 1.68e3T + 6.43e6T^{2} \)
29 \( 1 + 4.76e3T + 2.05e7T^{2} \)
31 \( 1 - 5.26e3T + 2.86e7T^{2} \)
37 \( 1 + 1.28e4T + 6.93e7T^{2} \)
41 \( 1 - 7.12e3T + 1.15e8T^{2} \)
43 \( 1 - 1.11e4T + 1.47e8T^{2} \)
47 \( 1 - 2.34e4T + 2.29e8T^{2} \)
53 \( 1 + 7.03e3T + 4.18e8T^{2} \)
59 \( 1 - 4.42e4T + 7.14e8T^{2} \)
61 \( 1 + 1.93e4T + 8.44e8T^{2} \)
67 \( 1 - 2.09e4T + 1.35e9T^{2} \)
71 \( 1 - 7.98e4T + 1.80e9T^{2} \)
73 \( 1 - 3.70e4T + 2.07e9T^{2} \)
79 \( 1 - 4.20e4T + 3.07e9T^{2} \)
83 \( 1 - 6.31e3T + 3.93e9T^{2} \)
89 \( 1 - 5.14e4T + 5.58e9T^{2} \)
97 \( 1 - 1.27e5T + 8.58e9T^{2} \)
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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

−10.92983293427933328364130222945, −10.16900144938232334814928508883, −9.299508776084385356618786635867, −7.80461893588277583545641870882, −6.71569989868823558255472362403, −5.71317237477042892511296477286, −5.19021090608180196970982237908, −3.73199070386786848316656292226, −2.33753260515470454110207922308, −1.05593541319359532572723104902, 1.05593541319359532572723104902, 2.33753260515470454110207922308, 3.73199070386786848316656292226, 5.19021090608180196970982237908, 5.71317237477042892511296477286, 6.71569989868823558255472362403, 7.80461893588277583545641870882, 9.299508776084385356618786635867, 10.16900144938232334814928508883, 10.92983293427933328364130222945

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