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Properties

Label 2-5e4-25.19-c1-0-30
Degree $2$
Conductor $625$
Sign $0.268 + 0.963i$
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  + (2.53 − 0.823i)2-s + (−0.446 − 0.614i)3-s + (4.12 − 2.99i)4-s + (−1.63 − 1.18i)6-s + 2.04i·7-s + (4.85 − 6.68i)8-s + (0.748 − 2.30i)9-s + (0.416 + 1.28i)11-s + (−3.68 − 1.19i)12-s + (1.25 + 0.407i)13-s + (1.68 + 5.17i)14-s + (3.65 − 11.2i)16-s + (−2.40 + 3.30i)17-s − 6.45i·18-s + (−3.95 − 2.87i)19-s + ⋯
L(s)  = 1  + (1.79 − 0.582i)2-s + (−0.257 − 0.354i)3-s + (2.06 − 1.49i)4-s + (−0.668 − 0.485i)6-s + 0.771i·7-s + (1.71 − 2.36i)8-s + (0.249 − 0.768i)9-s + (0.125 + 0.386i)11-s + (−1.06 − 0.345i)12-s + (0.348 + 0.113i)13-s + (0.449 + 1.38i)14-s + (0.913 − 2.81i)16-s + (−0.582 + 0.801i)17-s − 1.52i·18-s + (−0.906 − 0.658i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.268 + 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.268 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(625\)    =    \(5^{4}\)
Sign: $0.268 + 0.963i$
Analytic conductor: \(4.99065\)
Root analytic conductor: \(2.23397\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{625} (499, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 625,\ (\ :1/2),\ 0.268 + 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.10622 - 2.35775i\)
\(L(\frac12)\) \(\approx\) \(3.10622 - 2.35775i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.53 + 0.823i)T + (1.61 - 1.17i)T^{2} \)
3 \( 1 + (0.446 + 0.614i)T + (-0.927 + 2.85i)T^{2} \)
7 \( 1 - 2.04iT - 7T^{2} \)
11 \( 1 + (-0.416 - 1.28i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (-1.25 - 0.407i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (2.40 - 3.30i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (3.95 + 2.87i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-2.60 + 0.845i)T + (18.6 - 13.5i)T^{2} \)
29 \( 1 + (3.73 - 2.71i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (5.79 + 4.20i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-8.22 - 2.67i)T + (29.9 + 21.7i)T^{2} \)
41 \( 1 + (3.12 - 9.60i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 2.43iT - 43T^{2} \)
47 \( 1 + (-4.45 - 6.12i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (-0.365 - 0.502i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (-3.50 + 10.7i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-0.200 - 0.615i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (6.42 - 8.84i)T + (-20.7 - 63.7i)T^{2} \)
71 \( 1 + (-2.38 + 1.73i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (12.8 - 4.18i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (-1.52 + 1.11i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-1.39 + 1.92i)T + (-25.6 - 78.9i)T^{2} \)
89 \( 1 + (2.26 + 6.97i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-3.40 - 4.69i)T + (-29.9 + 92.2i)T^{2} \)
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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

−11.04844821399893742316239850866, −9.824591872619977209851802058035, −8.824322093286245214326929166753, −7.25552242597144936493823471807, −6.34474599439493835663127243396, −5.89818112705578441294527095854, −4.69468074937660770129931708772, −3.90090067311363221348236621851, −2.70086682241086125793755703813, −1.59907778057052912893534813283, 2.22591274258762815178925248537, 3.65649728063029700897946023909, 4.28546191597588531507493462250, 5.23694368640312235452403577786, 5.98535738577346764833108518053, 7.08517169082232936634268502956, 7.60444663479338396131847050512, 8.841118714000448516278531774433, 10.44351198070206533765844390353, 10.95645696024724518786852553573

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