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Properties

Label 2-425-17.16-c3-0-44
Degree $2$
Conductor $425$
Sign $-0.242 + 0.970i$
Analytic cond. $25.0758$
Root an. cond. $5.00757$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 8i·3-s − 7·4-s + 8i·6-s + 14i·7-s + 15·8-s − 37·9-s + 20i·11-s + 56i·12-s + 58·13-s − 14i·14-s + 41·16-s + (17 − 68i)17-s + 37·18-s + 80·19-s + ⋯
L(s)  = 1  − 0.353·2-s − 1.53i·3-s − 0.875·4-s + 0.544i·6-s + 0.755i·7-s + 0.662·8-s − 1.37·9-s + 0.548i·11-s + 1.34i·12-s + 1.23·13-s − 0.267i·14-s + 0.640·16-s + (0.242 − 0.970i)17-s + 0.484·18-s + 0.965·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.242 + 0.970i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.242 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(425\)    =    \(5^{2} \cdot 17\)
Sign: $-0.242 + 0.970i$
Analytic conductor: \(25.0758\)
Root analytic conductor: \(5.00757\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{425} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 425,\ (\ :3/2),\ -0.242 + 0.970i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.238114968\)
\(L(\frac12)\) \(\approx\) \(1.238114968\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
17 \( 1 + (-17 + 68i)T \)
good2 \( 1 + T + 8T^{2} \)
3 \( 1 + 8iT - 27T^{2} \)
7 \( 1 - 14iT - 343T^{2} \)
11 \( 1 - 20iT - 1.33e3T^{2} \)
13 \( 1 - 58T + 2.19e3T^{2} \)
19 \( 1 - 80T + 6.85e3T^{2} \)
23 \( 1 + 118iT - 1.21e4T^{2} \)
29 \( 1 - 126iT - 2.43e4T^{2} \)
31 \( 1 - 70iT - 2.97e4T^{2} \)
37 \( 1 - 134iT - 5.06e4T^{2} \)
41 \( 1 + 100iT - 6.89e4T^{2} \)
43 \( 1 + 272T + 7.95e4T^{2} \)
47 \( 1 - 464T + 1.03e5T^{2} \)
53 \( 1 + 642T + 1.48e5T^{2} \)
59 \( 1 + 180T + 2.05e5T^{2} \)
61 \( 1 - 110iT - 2.26e5T^{2} \)
67 \( 1 - 924T + 3.00e5T^{2} \)
71 \( 1 - 90iT - 3.57e5T^{2} \)
73 \( 1 + 828iT - 3.89e5T^{2} \)
79 \( 1 + 1.33e3iT - 4.93e5T^{2} \)
83 \( 1 + 552T + 5.71e5T^{2} \)
89 \( 1 - 1.49e3T + 7.04e5T^{2} \)
97 \( 1 + 1.37e3iT - 9.12e5T^{2} \)
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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

−10.46008187906253977505131536574, −9.270748603190843204829023757068, −8.613925334578762722110399497321, −7.79175214717372570323127820882, −6.91758270861956414710879069314, −5.87156800846396030581935525975, −4.83174077862578454627207069071, −3.15175839465597723304440587570, −1.70350788806466617721777604432, −0.64409132693716841096213039130, 1.01349883565595495735483123897, 3.61503695518772964617453264286, 3.88328923137175042867369871647, 5.08810310363216996046627288173, 5.98410985668681452200137246172, 7.70225044363267558479617561804, 8.534656107555749445681663459615, 9.385482675197433646936123831792, 10.01324110323010177715470449745, 10.77031262346374251875306629917

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