Properties

Label 2-245-1.1-c3-0-39
Degree $2$
Conductor $245$
Sign $-1$
Analytic cond. $14.4554$
Root an. cond. $3.80203$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.48·2-s − 0.850·3-s + 4.14·4-s − 5·5-s − 2.96·6-s − 13.4·8-s − 26.2·9-s − 17.4·10-s − 6.90·11-s − 3.52·12-s + 22.1·13-s + 4.25·15-s − 79.9·16-s − 88.3·17-s − 91.5·18-s − 36.9·19-s − 20.7·20-s − 24.0·22-s − 95.5·23-s + 11.4·24-s + 25·25-s + 77.1·26-s + 45.2·27-s + 269.·29-s + 14.8·30-s − 197.·31-s − 171.·32-s + ⋯
L(s)  = 1  + 1.23·2-s − 0.163·3-s + 0.518·4-s − 0.447·5-s − 0.201·6-s − 0.593·8-s − 0.973·9-s − 0.551·10-s − 0.189·11-s − 0.0848·12-s + 0.472·13-s + 0.0731·15-s − 1.24·16-s − 1.25·17-s − 1.19·18-s − 0.446·19-s − 0.231·20-s − 0.233·22-s − 0.866·23-s + 0.0970·24-s + 0.200·25-s + 0.582·26-s + 0.322·27-s + 1.72·29-s + 0.0901·30-s − 1.14·31-s − 0.946·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(245\)    =    \(5 \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(14.4554\)
Root analytic conductor: \(3.80203\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 245,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + 5T \)
7 \( 1 \)
good2 \( 1 - 3.48T + 8T^{2} \)
3 \( 1 + 0.850T + 27T^{2} \)
11 \( 1 + 6.90T + 1.33e3T^{2} \)
13 \( 1 - 22.1T + 2.19e3T^{2} \)
17 \( 1 + 88.3T + 4.91e3T^{2} \)
19 \( 1 + 36.9T + 6.85e3T^{2} \)
23 \( 1 + 95.5T + 1.21e4T^{2} \)
29 \( 1 - 269.T + 2.43e4T^{2} \)
31 \( 1 + 197.T + 2.97e4T^{2} \)
37 \( 1 - 2.14T + 5.06e4T^{2} \)
41 \( 1 + 174.T + 6.89e4T^{2} \)
43 \( 1 + 17.0T + 7.95e4T^{2} \)
47 \( 1 - 528.T + 1.03e5T^{2} \)
53 \( 1 + 641.T + 1.48e5T^{2} \)
59 \( 1 - 642.T + 2.05e5T^{2} \)
61 \( 1 + 142.T + 2.26e5T^{2} \)
67 \( 1 - 478.T + 3.00e5T^{2} \)
71 \( 1 - 105.T + 3.57e5T^{2} \)
73 \( 1 + 986.T + 3.89e5T^{2} \)
79 \( 1 + 1.09e3T + 4.93e5T^{2} \)
83 \( 1 - 1.23e3T + 5.71e5T^{2} \)
89 \( 1 - 711.T + 7.04e5T^{2} \)
97 \( 1 - 636.T + 9.12e5T^{2} \)
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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

−11.46727357399783508430770319764, −10.60995594645421670388143769536, −9.032197471101293627578520611794, −8.278910882999519101777007002571, −6.72071123553706145178269695424, −5.86873427559710920902523044208, −4.77916952741064063075795841995, −3.78336931782859209529655010140, −2.55146116296370456192564060613, 0, 2.55146116296370456192564060613, 3.78336931782859209529655010140, 4.77916952741064063075795841995, 5.86873427559710920902523044208, 6.72071123553706145178269695424, 8.278910882999519101777007002571, 9.032197471101293627578520611794, 10.60995594645421670388143769536, 11.46727357399783508430770319764

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