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Properties

Label 2-5e4-25.6-c1-0-4
Degree $2$
Conductor $625$
Sign $0.998 - 0.0627i$
Analytic cond. $4.99065$
Root an. cond. $2.23397$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.713 − 2.19i)2-s + (0.384 − 0.279i)3-s + (−2.69 + 1.95i)4-s + (−0.887 − 0.644i)6-s − 3.03·7-s + (2.48 + 1.80i)8-s + (−0.857 + 2.63i)9-s + (0.618 + 1.90i)11-s + (−0.489 + 1.50i)12-s + (0.441 − 1.35i)13-s + (2.16 + 6.66i)14-s + (0.136 − 0.420i)16-s + (1.50 + 1.09i)17-s + 6.40·18-s + (−0.730 − 0.530i)19-s + ⋯
L(s)  = 1  + (−0.504 − 1.55i)2-s + (0.221 − 0.161i)3-s + (−1.34 + 0.979i)4-s + (−0.362 − 0.263i)6-s − 1.14·7-s + (0.880 + 0.639i)8-s + (−0.285 + 0.879i)9-s + (0.186 + 0.573i)11-s + (−0.141 + 0.434i)12-s + (0.122 − 0.376i)13-s + (0.578 + 1.78i)14-s + (0.0341 − 0.105i)16-s + (0.365 + 0.265i)17-s + 1.51·18-s + (−0.167 − 0.121i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0627i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0627i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(625\)    =    \(5^{4}\)
Sign: $0.998 - 0.0627i$
Analytic conductor: \(4.99065\)
Root analytic conductor: \(2.23397\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{625} (126, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 625,\ (\ :1/2),\ 0.998 - 0.0627i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.596338 + 0.0187407i\)
\(L(\frac12)\) \(\approx\) \(0.596338 + 0.0187407i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
good2 \( 1 + (0.713 + 2.19i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (-0.384 + 0.279i)T + (0.927 - 2.85i)T^{2} \)
7 \( 1 + 3.03T + 7T^{2} \)
11 \( 1 + (-0.618 - 1.90i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (-0.441 + 1.35i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-1.50 - 1.09i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (0.730 + 0.530i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-1.02 - 3.16i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (3.20 - 2.32i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-5.21 - 3.78i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (1.18 - 3.63i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (0.566 - 1.74i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 3.59T + 43T^{2} \)
47 \( 1 + (3.88 - 2.82i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (7.68 - 5.58i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-3.28 + 10.1i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-4.41 - 13.5i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (8.64 + 6.28i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (10.0 - 7.32i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (0.0827 + 0.254i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-6.93 + 5.03i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (10.2 + 7.41i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (1.47 + 4.53i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (8.05 - 5.85i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.50241224257397284840976882660, −9.943279928854983457418878919515, −9.168672300875782641572355454524, −8.335862739684507776961413404040, −7.31254261228001964311882502128, −6.10135764910405274155832086908, −4.72513616523534872158595593619, −3.44237156230297242662585876078, −2.74392902853022881341437005232, −1.48417765416622127209427782830, 0.38723235570199566519346775603, 3.00019534410358030816140109604, 4.15042437662491425967491422160, 5.59382894793147770674556083883, 6.32673647371329236526848442478, 6.85714483758297597974021921694, 7.977383952205929165976296870740, 8.810759929499990106821755110419, 9.421041306267646508548504566427, 10.05352159186660096876441794660

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