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Properties

Label 2-725-145.129-c1-0-3
Degree $2$
Conductor $725$
Sign $-0.533 - 0.845i$
Analytic cond. $5.78915$
Root an. cond. $2.40606$
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  + (−1.20 − 0.581i)2-s + (−0.772 + 0.968i)3-s + (−0.127 − 0.159i)4-s + (1.49 − 0.720i)6-s + (3.02 + 2.40i)7-s + (0.657 + 2.87i)8-s + (0.325 + 1.42i)9-s + (−4.90 − 1.12i)11-s + 0.252·12-s + (3.13 + 0.715i)13-s + (−2.24 − 4.66i)14-s + (0.790 − 3.46i)16-s − 5.75·17-s + (0.436 − 1.91i)18-s + (3.76 − 3.00i)19-s + ⋯
L(s)  = 1  + (−0.853 − 0.411i)2-s + (−0.446 + 0.559i)3-s + (−0.0635 − 0.0796i)4-s + (0.610 − 0.294i)6-s + (1.14 + 0.910i)7-s + (0.232 + 1.01i)8-s + (0.108 + 0.475i)9-s + (−1.48 − 0.337i)11-s + 0.0729·12-s + (0.869 + 0.198i)13-s + (−0.600 − 1.24i)14-s + (0.197 − 0.865i)16-s − 1.39·17-s + (0.102 − 0.451i)18-s + (0.864 − 0.689i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.533 - 0.845i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.533 - 0.845i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(725\)    =    \(5^{2} \cdot 29\)
Sign: $-0.533 - 0.845i$
Analytic conductor: \(5.78915\)
Root analytic conductor: \(2.40606\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{725} (274, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 725,\ (\ :1/2),\ -0.533 - 0.845i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.230803 + 0.418687i\)
\(L(\frac12)\) \(\approx\) \(0.230803 + 0.418687i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
29 \( 1 + (2.84 + 4.57i)T \)
good2 \( 1 + (1.20 + 0.581i)T + (1.24 + 1.56i)T^{2} \)
3 \( 1 + (0.772 - 0.968i)T + (-0.667 - 2.92i)T^{2} \)
7 \( 1 + (-3.02 - 2.40i)T + (1.55 + 6.82i)T^{2} \)
11 \( 1 + (4.90 + 1.12i)T + (9.91 + 4.77i)T^{2} \)
13 \( 1 + (-3.13 - 0.715i)T + (11.7 + 5.64i)T^{2} \)
17 \( 1 + 5.75T + 17T^{2} \)
19 \( 1 + (-3.76 + 3.00i)T + (4.22 - 18.5i)T^{2} \)
23 \( 1 + (-1.51 - 3.14i)T + (-14.3 + 17.9i)T^{2} \)
31 \( 1 + (1.34 - 2.79i)T + (-19.3 - 24.2i)T^{2} \)
37 \( 1 + (-0.603 - 2.64i)T + (-33.3 + 16.0i)T^{2} \)
41 \( 1 - 12.5iT - 41T^{2} \)
43 \( 1 + (7.79 - 3.75i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 + (-0.0495 + 0.217i)T + (-42.3 - 20.3i)T^{2} \)
53 \( 1 + (2.90 - 6.02i)T + (-33.0 - 41.4i)T^{2} \)
59 \( 1 - 4.63T + 59T^{2} \)
61 \( 1 + (7.39 + 5.89i)T + (13.5 + 59.4i)T^{2} \)
67 \( 1 + (12.6 - 2.88i)T + (60.3 - 29.0i)T^{2} \)
71 \( 1 + (-1.66 + 7.31i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (3.18 - 1.53i)T + (45.5 - 57.0i)T^{2} \)
79 \( 1 + (-4.33 + 0.990i)T + (71.1 - 34.2i)T^{2} \)
83 \( 1 + (6.59 - 5.25i)T + (18.4 - 80.9i)T^{2} \)
89 \( 1 + (4.20 - 8.72i)T + (-55.4 - 69.5i)T^{2} \)
97 \( 1 + (-9.77 - 12.2i)T + (-21.5 + 94.5i)T^{2} \)
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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

−10.89491884050977451206876511238, −9.866128805140223791549242624775, −9.054194319701959669812656497564, −8.275313522167306076169857238161, −7.70260756103694024540688932269, −5.99771578883302895567790801423, −5.08393889845750727729506562641, −4.72133641427073591986494853358, −2.72516740781264556906578848769, −1.64075363209467797606092223192, 0.35994351928134128113001273401, 1.67896455743213316265994888946, 3.62786362807992411296977182493, 4.68027759164791548637347306188, 5.81590136252529780383328009433, 7.10208375704415276595381511661, 7.38534513199581272632448160939, 8.310423903515609788903830893294, 8.988818620994784313650639363724, 10.22452491896622596603477834907

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