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Properties

Label 2-245-5.4-c5-0-88
Degree $2$
Conductor $245$
Sign $-0.804 - 0.593i$
Analytic cond. $39.2940$
Root an. cond. $6.26849$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 6.63i·2-s − 19.8i·3-s − 12·4-s + (45 + 33.1i)5-s − 132·6-s − 132. i·8-s − 153·9-s + (220. − 298. i)10-s + 252·11-s + 238. i·12-s − 119. i·13-s + (660 − 895. i)15-s − 1.26e3·16-s − 689. i·17-s + 1.01e3i·18-s + 220·19-s + ⋯
L(s)  = 1  − 1.17i·2-s − 1.27i·3-s − 0.375·4-s + (0.804 + 0.593i)5-s − 1.49·6-s − 0.732i·8-s − 0.629·9-s + (0.695 − 0.943i)10-s + 0.627·11-s + 0.478i·12-s − 0.195i·13-s + (0.757 − 1.02i)15-s − 1.23·16-s − 0.578i·17-s + 0.738i·18-s + 0.139·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.804 - 0.593i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.804 - 0.593i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(245\)    =    \(5 \cdot 7^{2}\)
Sign: $-0.804 - 0.593i$
Analytic conductor: \(39.2940\)
Root analytic conductor: \(6.26849\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{245} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 245,\ (\ :5/2),\ -0.804 - 0.593i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.244685329\)
\(L(\frac12)\) \(\approx\) \(2.244685329\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-45 - 33.1i)T \)
7 \( 1 \)
good2 \( 1 + 6.63iT - 32T^{2} \)
3 \( 1 + 19.8iT - 243T^{2} \)
11 \( 1 - 252T + 1.61e5T^{2} \)
13 \( 1 + 119. iT - 3.71e5T^{2} \)
17 \( 1 + 689. iT - 1.41e6T^{2} \)
19 \( 1 - 220T + 2.47e6T^{2} \)
23 \( 1 + 2.43e3iT - 6.43e6T^{2} \)
29 \( 1 + 6.93e3T + 2.05e7T^{2} \)
31 \( 1 + 6.75e3T + 2.86e7T^{2} \)
37 \( 1 + 1.39e4iT - 6.93e7T^{2} \)
41 \( 1 - 198T + 1.15e8T^{2} \)
43 \( 1 - 417. iT - 1.47e8T^{2} \)
47 \( 1 + 1.05e4iT - 2.29e8T^{2} \)
53 \( 1 - 5.82e3iT - 4.18e8T^{2} \)
59 \( 1 - 2.46e4T + 7.14e8T^{2} \)
61 \( 1 - 5.69e3T + 8.44e8T^{2} \)
67 \( 1 - 4.36e4iT - 1.35e9T^{2} \)
71 \( 1 - 5.33e4T + 1.80e9T^{2} \)
73 \( 1 - 7.09e4iT - 2.07e9T^{2} \)
79 \( 1 - 5.19e4T + 3.07e9T^{2} \)
83 \( 1 + 6.18e4iT - 3.93e9T^{2} \)
89 \( 1 - 9.99e3T + 5.58e9T^{2} \)
97 \( 1 + 1.01e5iT - 8.58e9T^{2} \)
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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

−10.89878479553880040152770734111, −9.872722327872960041377385898405, −8.990206537984244331858018170330, −7.35837584168313849192961712444, −6.79333607068090278460773611092, −5.68701279270296160301571207986, −3.76229697439406926303936169665, −2.45071780976177305158568266000, −1.77659665108182358239221945194, −0.60659669344056432807751757857, 1.76357753476231496978551879186, 3.69624693012619572428665713867, 4.90495009888309641192672677415, 5.62008010739812713787098889479, 6.62287456979023731019525533561, 7.907116925196730367800649140759, 9.084618099219475019334436097832, 9.501933812801325290737587307117, 10.66938773720813264628867435279, 11.61492400488104118989051106695

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