Properties

Label 2-245-1.1-c5-0-24
Degree $2$
Conductor $245$
Sign $-1$
Analytic cond. $39.2940$
Root an. cond. $6.26849$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 7.22·2-s − 15.9·3-s + 20.2·4-s − 25·5-s + 115.·6-s + 85.0·8-s + 10.4·9-s + 180.·10-s − 387.·11-s − 322.·12-s − 920.·13-s + 398.·15-s − 1.26e3·16-s + 678.·17-s − 75.7·18-s + 2.76e3·19-s − 505.·20-s + 2.79e3·22-s + 2.06e3·23-s − 1.35e3·24-s + 625·25-s + 6.65e3·26-s + 3.70e3·27-s − 2.92e3·29-s − 2.87e3·30-s + 4.25e3·31-s + 6.40e3·32-s + ⋯
L(s)  = 1  − 1.27·2-s − 1.02·3-s + 0.632·4-s − 0.447·5-s + 1.30·6-s + 0.469·8-s + 0.0431·9-s + 0.571·10-s − 0.964·11-s − 0.645·12-s − 1.51·13-s + 0.456·15-s − 1.23·16-s + 0.569·17-s − 0.0550·18-s + 1.75·19-s − 0.282·20-s + 1.23·22-s + 0.814·23-s − 0.479·24-s + 0.200·25-s + 1.93·26-s + 0.977·27-s − 0.645·29-s − 0.583·30-s + 0.795·31-s + 1.10·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(245\)    =    \(5 \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(39.2940\)
Root analytic conductor: \(6.26849\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{245} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 245,\ (\ :5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + 25T \)
7 \( 1 \)
good2 \( 1 + 7.22T + 32T^{2} \)
3 \( 1 + 15.9T + 243T^{2} \)
11 \( 1 + 387.T + 1.61e5T^{2} \)
13 \( 1 + 920.T + 3.71e5T^{2} \)
17 \( 1 - 678.T + 1.41e6T^{2} \)
19 \( 1 - 2.76e3T + 2.47e6T^{2} \)
23 \( 1 - 2.06e3T + 6.43e6T^{2} \)
29 \( 1 + 2.92e3T + 2.05e7T^{2} \)
31 \( 1 - 4.25e3T + 2.86e7T^{2} \)
37 \( 1 + 154.T + 6.93e7T^{2} \)
41 \( 1 + 7.60e3T + 1.15e8T^{2} \)
43 \( 1 - 1.35e4T + 1.47e8T^{2} \)
47 \( 1 - 1.50e4T + 2.29e8T^{2} \)
53 \( 1 - 1.80e4T + 4.18e8T^{2} \)
59 \( 1 - 6.47e3T + 7.14e8T^{2} \)
61 \( 1 - 4.00e3T + 8.44e8T^{2} \)
67 \( 1 + 3.36e4T + 1.35e9T^{2} \)
71 \( 1 + 6.14e4T + 1.80e9T^{2} \)
73 \( 1 + 4.26e4T + 2.07e9T^{2} \)
79 \( 1 + 1.07e5T + 3.07e9T^{2} \)
83 \( 1 - 9.90e4T + 3.93e9T^{2} \)
89 \( 1 - 7.66e3T + 5.58e9T^{2} \)
97 \( 1 - 8.70e4T + 8.58e9T^{2} \)
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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

−10.54922614687735474133115350162, −9.927446489189857789710214167547, −8.888711862044523054936002580983, −7.63242626740884179345743680725, −7.23301421550558077010934328792, −5.55496298657086659619928415561, −4.76439962533707293214923117743, −2.77393816521697384147327131406, −0.936751823763627328388798543442, 0, 0.936751823763627328388798543442, 2.77393816521697384147327131406, 4.76439962533707293214923117743, 5.55496298657086659619928415561, 7.23301421550558077010934328792, 7.63242626740884179345743680725, 8.888711862044523054936002580983, 9.927446489189857789710214167547, 10.54922614687735474133115350162

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