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Properties

Label 2-5e4-125.104-c1-0-0
Degree $2$
Conductor $625$
Sign $0.0302 - 0.999i$
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  + (−0.475 − 1.85i)2-s + (0.722 − 0.137i)3-s + (−1.44 + 0.795i)4-s + (−0.598 − 1.27i)6-s + (−2.04 + 2.81i)7-s + (−0.456 − 0.486i)8-s + (−2.28 + 0.905i)9-s + (−1.60 + 0.412i)11-s + (−0.935 + 0.773i)12-s + (−1.71 − 4.34i)13-s + (6.17 + 2.44i)14-s + (−2.45 + 3.86i)16-s + (1.70 − 3.09i)17-s + (2.76 + 3.80i)18-s + (−1.54 + 8.08i)19-s + ⋯
L(s)  = 1  + (−0.336 − 1.30i)2-s + (0.417 − 0.0795i)3-s + (−0.723 + 0.397i)4-s + (−0.244 − 0.519i)6-s + (−0.771 + 1.06i)7-s + (−0.161 − 0.171i)8-s + (−0.762 + 0.301i)9-s + (−0.484 + 0.124i)11-s + (−0.270 + 0.223i)12-s + (−0.476 − 1.20i)13-s + (1.64 + 0.653i)14-s + (−0.613 + 0.965i)16-s + (0.412 − 0.751i)17-s + (0.650 + 0.895i)18-s + (−0.353 + 1.85i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0302 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0302 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(625\)    =    \(5^{4}\)
Sign: $0.0302 - 0.999i$
Analytic conductor: \(4.99065\)
Root analytic conductor: \(2.23397\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{625} (274, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 625,\ (\ :1/2),\ 0.0302 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0401350 + 0.0389384i\)
\(L(\frac12)\) \(\approx\) \(0.0401350 + 0.0389384i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.475 + 1.85i)T + (-1.75 + 0.963i)T^{2} \)
3 \( 1 + (-0.722 + 0.137i)T + (2.78 - 1.10i)T^{2} \)
7 \( 1 + (2.04 - 2.81i)T + (-2.16 - 6.65i)T^{2} \)
11 \( 1 + (1.60 - 0.412i)T + (9.63 - 5.29i)T^{2} \)
13 \( 1 + (1.71 + 4.34i)T + (-9.47 + 8.89i)T^{2} \)
17 \( 1 + (-1.70 + 3.09i)T + (-9.10 - 14.3i)T^{2} \)
19 \( 1 + (1.54 - 8.08i)T + (-17.6 - 6.99i)T^{2} \)
23 \( 1 + (5.62 + 0.354i)T + (22.8 + 2.88i)T^{2} \)
29 \( 1 + (2.76 + 0.349i)T + (28.0 + 7.21i)T^{2} \)
31 \( 1 + (-3.57 - 1.96i)T + (16.6 + 26.1i)T^{2} \)
37 \( 1 + (3.80 + 2.41i)T + (15.7 + 33.4i)T^{2} \)
41 \( 1 + (0.634 + 10.0i)T + (-40.6 + 5.13i)T^{2} \)
43 \( 1 + (0.711 - 0.231i)T + (34.7 - 25.2i)T^{2} \)
47 \( 1 + (4.35 - 4.63i)T + (-2.95 - 46.9i)T^{2} \)
53 \( 1 + (-6.58 - 3.10i)T + (33.7 + 40.8i)T^{2} \)
59 \( 1 + (2.95 + 3.57i)T + (-11.0 + 57.9i)T^{2} \)
61 \( 1 + (0.259 - 4.12i)T + (-60.5 - 7.64i)T^{2} \)
67 \( 1 + (0.287 + 2.27i)T + (-64.8 + 16.6i)T^{2} \)
71 \( 1 + (-5.95 - 5.59i)T + (4.45 + 70.8i)T^{2} \)
73 \( 1 + (3.49 + 2.89i)T + (13.6 + 71.7i)T^{2} \)
79 \( 1 + (1.69 + 8.88i)T + (-73.4 + 29.0i)T^{2} \)
83 \( 1 + (-8.24 - 1.57i)T + (77.1 + 30.5i)T^{2} \)
89 \( 1 + (-8.05 + 9.73i)T + (-16.6 - 87.4i)T^{2} \)
97 \( 1 + (0.599 - 4.74i)T + (-93.9 - 24.1i)T^{2} \)
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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

−10.58400390926415006591504492800, −10.10777509070081691797806800306, −9.300104546865417091127250016533, −8.447583862306777569860556453422, −7.67853134390749480061178630659, −6.07735928342953420153134478115, −5.41450501836359410671122273531, −3.62091345349905550910123597958, −2.83963383011657433484057884777, −2.07195317157707568913270569337, 0.02919123142832096940114833837, 2.57942536961931286497188249326, 3.85031137266659100462143494427, 5.03983854258745892600647565323, 6.31208408993801830569222027932, 6.75944314374168808528384038512, 7.71691627460849515063549163223, 8.477623586747958091034289727034, 9.322930621033128016983714972620, 10.00974926426135199923036519290

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