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Properties

Label 2-245-5.4-c5-0-9
Degree $2$
Conductor $245$
Sign $0.0831 - 0.996i$
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  − 8.00i·2-s + 7.64i·3-s − 32.1·4-s + (−4.64 + 55.7i)5-s + 61.1·6-s + 1.27i·8-s + 184.·9-s + (446. + 37.2i)10-s − 129.·11-s − 245. i·12-s + 1.05e3i·13-s + (−425. − 35.5i)15-s − 1.01e3·16-s − 1.17e3i·17-s − 1.47e3i·18-s + 520.·19-s + ⋯
L(s)  = 1  − 1.41i·2-s + 0.490i·3-s − 1.00·4-s + (−0.0831 + 0.996i)5-s + 0.693·6-s + 0.00704i·8-s + 0.759·9-s + (1.41 + 0.117i)10-s − 0.323·11-s − 0.492i·12-s + 1.72i·13-s + (−0.488 − 0.0407i)15-s − 0.994·16-s − 0.982i·17-s − 1.07i·18-s + 0.330·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0831 - 0.996i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.0831 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(245\)    =    \(5 \cdot 7^{2}\)
Sign: $0.0831 - 0.996i$
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: \((2,\ 245,\ (\ :5/2),\ 0.0831 - 0.996i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.7804916620\)
\(L(\frac12)\) \(\approx\) \(0.7804916620\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (4.64 - 55.7i)T \)
7 \( 1 \)
good2 \( 1 + 8.00iT - 32T^{2} \)
3 \( 1 - 7.64iT - 243T^{2} \)
11 \( 1 + 129.T + 1.61e5T^{2} \)
13 \( 1 - 1.05e3iT - 3.71e5T^{2} \)
17 \( 1 + 1.17e3iT - 1.41e6T^{2} \)
19 \( 1 - 520.T + 2.47e6T^{2} \)
23 \( 1 + 3.16e3iT - 6.43e6T^{2} \)
29 \( 1 + 5.58e3T + 2.05e7T^{2} \)
31 \( 1 + 6.89e3T + 2.86e7T^{2} \)
37 \( 1 - 8.79e3iT - 6.93e7T^{2} \)
41 \( 1 + 896.T + 1.15e8T^{2} \)
43 \( 1 - 1.12e4iT - 1.47e8T^{2} \)
47 \( 1 - 1.63e4iT - 2.29e8T^{2} \)
53 \( 1 - 324. iT - 4.18e8T^{2} \)
59 \( 1 - 45.7T + 7.14e8T^{2} \)
61 \( 1 + 3.19e4T + 8.44e8T^{2} \)
67 \( 1 - 7.84e3iT - 1.35e9T^{2} \)
71 \( 1 - 2.77e3T + 1.80e9T^{2} \)
73 \( 1 + 2.58e4iT - 2.07e9T^{2} \)
79 \( 1 - 1.58e3T + 3.07e9T^{2} \)
83 \( 1 + 1.98e4iT - 3.93e9T^{2} \)
89 \( 1 + 8.30e4T + 5.58e9T^{2} \)
97 \( 1 - 8.38e4iT - 8.58e9T^{2} \)
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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

−11.26436651861922128364121630759, −10.70623907662040344851912433001, −9.718004276209682320253021594970, −9.215736260469098546680185638648, −7.39832631154315012141027609609, −6.56456727375191404559704218287, −4.69090035891257823203801027721, −3.82369371584694472103935884414, −2.71120221075649181004932312755, −1.61445727273341861993210380311, 0.21416519921534665718670877820, 1.71968851275402058479477783741, 3.85466362510501126609183546113, 5.34098373985083861924640221570, 5.74655294962347256692933548809, 7.28271185923759429033677086382, 7.75732447475879692746427449731, 8.628956137766294709983991082773, 9.723704785380275474321214397509, 10.97301117610061957398074251680

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