Properties

Label 2-245-1.1-c3-0-34
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  + 0.644·2-s + 4.18·3-s − 7.58·4-s + 5·5-s + 2.69·6-s − 10.0·8-s − 9.47·9-s + 3.22·10-s − 47.7·11-s − 31.7·12-s − 57.2·13-s + 20.9·15-s + 54.1·16-s + 36.9·17-s − 6.10·18-s − 30.7·19-s − 37.9·20-s − 30.7·22-s + 53.1·23-s − 42.0·24-s + 25·25-s − 36.8·26-s − 152.·27-s − 195.·29-s + 13.4·30-s − 257.·31-s + 115.·32-s + ⋯
L(s)  = 1  + 0.227·2-s + 0.805·3-s − 0.948·4-s + 0.447·5-s + 0.183·6-s − 0.443·8-s − 0.350·9-s + 0.101·10-s − 1.30·11-s − 0.763·12-s − 1.22·13-s + 0.360·15-s + 0.846·16-s + 0.527·17-s − 0.0799·18-s − 0.371·19-s − 0.423·20-s − 0.298·22-s + 0.481·23-s − 0.357·24-s + 0.200·25-s − 0.278·26-s − 1.08·27-s − 1.25·29-s + 0.0821·30-s − 1.49·31-s + 0.637·32-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 \( 1 - 0.644T + 8T^{2} \)
3 \( 1 - 4.18T + 27T^{2} \)
11 \( 1 + 47.7T + 1.33e3T^{2} \)
13 \( 1 + 57.2T + 2.19e3T^{2} \)
17 \( 1 - 36.9T + 4.91e3T^{2} \)
19 \( 1 + 30.7T + 6.85e3T^{2} \)
23 \( 1 - 53.1T + 1.21e4T^{2} \)
29 \( 1 + 195.T + 2.43e4T^{2} \)
31 \( 1 + 257.T + 2.97e4T^{2} \)
37 \( 1 - 346.T + 5.06e4T^{2} \)
41 \( 1 + 267.T + 6.89e4T^{2} \)
43 \( 1 + 176.T + 7.95e4T^{2} \)
47 \( 1 - 311.T + 1.03e5T^{2} \)
53 \( 1 + 492.T + 1.48e5T^{2} \)
59 \( 1 + 98.7T + 2.05e5T^{2} \)
61 \( 1 + 82.1T + 2.26e5T^{2} \)
67 \( 1 - 654.T + 3.00e5T^{2} \)
71 \( 1 - 779.T + 3.57e5T^{2} \)
73 \( 1 - 829.T + 3.89e5T^{2} \)
79 \( 1 + 769.T + 4.93e5T^{2} \)
83 \( 1 + 613.T + 5.71e5T^{2} \)
89 \( 1 - 457.T + 7.04e5T^{2} \)
97 \( 1 - 1.41e3T + 9.12e5T^{2} \)
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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

−11.08642909721367743157944652558, −9.882418110663964405906443525118, −9.304500247162222158705007652619, −8.261989211356264678898520734710, −7.47304804001997634684633106788, −5.68944064496039057683454108753, −4.94469303529271725817132153376, −3.45221942359868514676449935384, −2.33418622646846625690068666008, 0, 2.33418622646846625690068666008, 3.45221942359868514676449935384, 4.94469303529271725817132153376, 5.68944064496039057683454108753, 7.47304804001997634684633106788, 8.261989211356264678898520734710, 9.304500247162222158705007652619, 9.882418110663964405906443525118, 11.08642909721367743157944652558

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