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Properties

Label 2-5e4-25.6-c1-0-22
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  + (−0.350 − 1.07i)2-s + (−2.10 + 1.52i)3-s + (0.578 − 0.420i)4-s + (2.38 + 1.73i)6-s − 0.407·7-s + (−2.48 − 1.80i)8-s + (1.16 − 3.58i)9-s + (0.618 + 1.90i)11-s + (−0.574 + 1.76i)12-s + (0.216 − 0.666i)13-s + (0.142 + 0.439i)14-s + (−0.636 + 1.95i)16-s + (1.28 + 0.930i)17-s − 4.27·18-s + (−4.00 − 2.90i)19-s + ⋯
L(s)  = 1  + (−0.247 − 0.762i)2-s + (−1.21 + 0.883i)3-s + (0.289 − 0.210i)4-s + (0.974 + 0.708i)6-s − 0.153·7-s + (−0.880 − 0.639i)8-s + (0.388 − 1.19i)9-s + (0.186 + 0.573i)11-s + (−0.165 + 0.510i)12-s + (0.0600 − 0.184i)13-s + (0.0381 + 0.117i)14-s + (−0.159 + 0.489i)16-s + (0.310 + 0.225i)17-s − 1.00·18-s + (−0.918 − 0.667i)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

\(L(1)\) \(\approx\) \(0.00804481 - 0.255990i\)
\(L(\frac12)\) \(\approx\) \(0.00804481 - 0.255990i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.350 + 1.07i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (2.10 - 1.52i)T + (0.927 - 2.85i)T^{2} \)
7 \( 1 + 0.407T + 7T^{2} \)
11 \( 1 + (-0.618 - 1.90i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (-0.216 + 0.666i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-1.28 - 0.930i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (4.00 + 2.90i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (0.371 + 1.14i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (4.45 - 3.23i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (6.63 + 4.82i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-1.58 + 4.88i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-2.22 + 6.86i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 9.16T + 43T^{2} \)
47 \( 1 + (1.03 - 0.748i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (4.10 - 2.98i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (2.00 - 6.18i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (2.91 + 8.95i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (2.49 + 1.81i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (5.55 - 4.03i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (0.168 + 0.518i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (4.43 - 3.22i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-0.788 - 0.572i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (0.700 + 2.15i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-12.3 + 8.94i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.38164861755505387952560434696, −9.675460921412399652990152309406, −8.931860825345804115221371759755, −7.33189910561599451435662617361, −6.32728982480998095046106399625, −5.61386594392316720966734266150, −4.53379710974685949418202268265, −3.48950638585481554116063338516, −1.94674063366936413467718186619, −0.16755164771674886640333842349, 1.70988468391073702185303829647, 3.37115871639013796204886702823, 5.06311192576698538965892973391, 6.08083159639752113651504515085, 6.40877112238329463365039992668, 7.35932136710286928419347902211, 8.071754613203389418301106140611, 9.057418275935666252102214657310, 10.31644030438435909163315060774, 11.41102548809869655568115745432

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