Properties

Label 2-731-1.1-c1-0-17
Degree $2$
Conductor $731$
Sign $1$
Analytic cond. $5.83706$
Root an. cond. $2.41600$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.148·2-s + 0.359·3-s − 1.97·4-s + 2.35·5-s + 0.0533·6-s − 1.55·7-s − 0.590·8-s − 2.87·9-s + 0.350·10-s + 1.82·11-s − 0.710·12-s + 5.22·13-s − 0.231·14-s + 0.847·15-s + 3.86·16-s + 17-s − 0.426·18-s + 3.40·19-s − 4.66·20-s − 0.560·21-s + 0.270·22-s + 5.17·23-s − 0.212·24-s + 0.568·25-s + 0.776·26-s − 2.10·27-s + 3.08·28-s + ⋯
L(s)  = 1  + 0.104·2-s + 0.207·3-s − 0.988·4-s + 1.05·5-s + 0.0217·6-s − 0.589·7-s − 0.208·8-s − 0.956·9-s + 0.110·10-s + 0.549·11-s − 0.205·12-s + 1.45·13-s − 0.0618·14-s + 0.218·15-s + 0.967·16-s + 0.242·17-s − 0.100·18-s + 0.782·19-s − 1.04·20-s − 0.122·21-s + 0.0576·22-s + 1.07·23-s − 0.0433·24-s + 0.113·25-s + 0.152·26-s − 0.405·27-s + 0.583·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $1$
Analytic conductor: \(5.83706\)
Root analytic conductor: \(2.41600\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 731,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.600442975\)
\(L(\frac12)\) \(\approx\) \(1.600442975\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 - T \)
43 \( 1 + T \)
good2 \( 1 - 0.148T + 2T^{2} \)
3 \( 1 - 0.359T + 3T^{2} \)
5 \( 1 - 2.35T + 5T^{2} \)
7 \( 1 + 1.55T + 7T^{2} \)
11 \( 1 - 1.82T + 11T^{2} \)
13 \( 1 - 5.22T + 13T^{2} \)
19 \( 1 - 3.40T + 19T^{2} \)
23 \( 1 - 5.17T + 23T^{2} \)
29 \( 1 - 7.10T + 29T^{2} \)
31 \( 1 - 0.110T + 31T^{2} \)
37 \( 1 + 0.00648T + 37T^{2} \)
41 \( 1 - 3.74T + 41T^{2} \)
47 \( 1 + 5.87T + 47T^{2} \)
53 \( 1 - 7.44T + 53T^{2} \)
59 \( 1 - 8.85T + 59T^{2} \)
61 \( 1 + 9.60T + 61T^{2} \)
67 \( 1 - 7.29T + 67T^{2} \)
71 \( 1 + 7.51T + 71T^{2} \)
73 \( 1 - 12.3T + 73T^{2} \)
79 \( 1 + 3.82T + 79T^{2} \)
83 \( 1 + 8.31T + 83T^{2} \)
89 \( 1 - 15.3T + 89T^{2} \)
97 \( 1 + 3.92T + 97T^{2} \)
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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

−10.15188748705249181812421274063, −9.402237086058335509088361652404, −8.867865597907231502701650645625, −8.113612036102349532175561591114, −6.59006539091929379240388380186, −5.88926978949063443746496724181, −5.11910518295871505092783908580, −3.75438767142216963958400552087, −2.90987016037074983401332351125, −1.12656838359313271332800992460, 1.12656838359313271332800992460, 2.90987016037074983401332351125, 3.75438767142216963958400552087, 5.11910518295871505092783908580, 5.88926978949063443746496724181, 6.59006539091929379240388380186, 8.113612036102349532175561591114, 8.867865597907231502701650645625, 9.402237086058335509088361652404, 10.15188748705249181812421274063

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