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Properties

Label 2-5e4-25.11-c1-0-14
Degree $2$
Conductor $625$
Sign $-0.0209 - 0.999i$
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  + (0.264 − 0.192i)2-s + (−0.530 + 1.63i)3-s + (−0.585 + 1.80i)4-s + (0.173 + 0.533i)6-s + 3.42·7-s + (0.393 + 1.20i)8-s + (0.0465 + 0.0337i)9-s + (4.32 − 3.13i)11-s + (−2.62 − 1.90i)12-s + (2.84 + 2.06i)13-s + (0.905 − 0.657i)14-s + (−2.72 − 1.98i)16-s + (0.790 + 2.43i)17-s + 0.0187·18-s + (−0.626 − 1.92i)19-s + ⋯
L(s)  = 1  + (0.186 − 0.135i)2-s + (−0.306 + 0.941i)3-s + (−0.292 + 0.900i)4-s + (0.0707 + 0.217i)6-s + 1.29·7-s + (0.138 + 0.427i)8-s + (0.0155 + 0.0112i)9-s + (1.30 − 0.946i)11-s + (−0.758 − 0.551i)12-s + (0.790 + 0.574i)13-s + (0.241 − 0.175i)14-s + (−0.681 − 0.495i)16-s + (0.191 + 0.590i)17-s + 0.00442·18-s + (−0.143 − 0.442i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0209 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0209 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(625\)    =    \(5^{4}\)
Sign: $-0.0209 - 0.999i$
Analytic conductor: \(4.99065\)
Root analytic conductor: \(2.23397\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{625} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 625,\ (\ :1/2),\ -0.0209 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.20379 + 1.22927i\)
\(L(\frac12)\) \(\approx\) \(1.20379 + 1.22927i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.264 + 0.192i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 + (0.530 - 1.63i)T + (-2.42 - 1.76i)T^{2} \)
7 \( 1 - 3.42T + 7T^{2} \)
11 \( 1 + (-4.32 + 3.13i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (-2.84 - 2.06i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-0.790 - 2.43i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (0.626 + 1.92i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (6.12 - 4.45i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (-1.46 + 4.51i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-0.501 - 1.54i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-0.0108 - 0.00789i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (7.82 + 5.68i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 2.32T + 43T^{2} \)
47 \( 1 + (-2.14 + 6.60i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-0.532 + 1.63i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-0.0179 - 0.0130i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (3.16 - 2.30i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-1.27 - 3.91i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (-0.722 + 2.22i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (1.22 - 0.889i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-0.131 + 0.405i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (1.86 + 5.74i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (-4.92 + 3.58i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (4.94 - 15.2i)T + (-78.4 - 57.0i)T^{2} \)
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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

−11.09687554723623914411725434312, −10.03408903207424725688695489173, −8.898281482452368385873496422150, −8.457165210430752825299125624091, −7.45835072544237433755939259781, −6.14273196879071052088557341656, −5.06072709714584158884728905436, −4.06951278376580550051742627236, −3.69925496251286887792335774235, −1.75613798168452240624398295151, 1.12540472155750276830490618914, 1.83753505614248751085069420760, 4.05910176359013003771456194072, 4.87799658141472860596787887484, 6.02082727548166072669129919286, 6.63725112278586040961607028072, 7.62352446805756792786847260680, 8.531279557980649761859935482341, 9.580091587699128259042795534666, 10.42051924040532690520221968961

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