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Properties

Label 2-725-145.129-c1-0-5
Degree $2$
Conductor $725$
Sign $0.560 - 0.828i$
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  + (−1.16 − 0.561i)2-s + (−1.90 + 2.38i)3-s + (−0.201 − 0.252i)4-s + (3.56 − 1.71i)6-s + (−2.58 − 2.06i)7-s + (0.669 + 2.93i)8-s + (−1.40 − 6.16i)9-s + (−2.84 − 0.649i)11-s + 0.984·12-s + (3.20 + 0.731i)13-s + (1.85 + 3.85i)14-s + (0.723 − 3.16i)16-s − 2.25·17-s + (−1.82 + 7.98i)18-s + (3.50 − 2.79i)19-s + ⋯
L(s)  = 1  + (−0.825 − 0.397i)2-s + (−1.09 + 1.37i)3-s + (−0.100 − 0.126i)4-s + (1.45 − 0.700i)6-s + (−0.976 − 0.778i)7-s + (0.236 + 1.03i)8-s + (−0.468 − 2.05i)9-s + (−0.857 − 0.195i)11-s + 0.284·12-s + (0.889 + 0.202i)13-s + (0.496 + 1.03i)14-s + (0.180 − 0.792i)16-s − 0.547·17-s + (−0.429 + 1.88i)18-s + (0.803 − 0.640i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.560 - 0.828i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.560 - 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(725\)    =    \(5^{2} \cdot 29\)
Sign: $0.560 - 0.828i$
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: \((2,\ 725,\ (\ :1/2),\ 0.560 - 0.828i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.292661 + 0.155372i\)
\(L(\frac12)\) \(\approx\) \(0.292661 + 0.155372i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.72 - 5.10i)T \)
good2 \( 1 + (1.16 + 0.561i)T + (1.24 + 1.56i)T^{2} \)
3 \( 1 + (1.90 - 2.38i)T + (-0.667 - 2.92i)T^{2} \)
7 \( 1 + (2.58 + 2.06i)T + (1.55 + 6.82i)T^{2} \)
11 \( 1 + (2.84 + 0.649i)T + (9.91 + 4.77i)T^{2} \)
13 \( 1 + (-3.20 - 0.731i)T + (11.7 + 5.64i)T^{2} \)
17 \( 1 + 2.25T + 17T^{2} \)
19 \( 1 + (-3.50 + 2.79i)T + (4.22 - 18.5i)T^{2} \)
23 \( 1 + (3.83 + 7.95i)T + (-14.3 + 17.9i)T^{2} \)
31 \( 1 + (2.38 - 4.95i)T + (-19.3 - 24.2i)T^{2} \)
37 \( 1 + (-0.970 - 4.25i)T + (-33.3 + 16.0i)T^{2} \)
41 \( 1 - 1.22iT - 41T^{2} \)
43 \( 1 + (-0.197 + 0.0952i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 + (0.906 - 3.97i)T + (-42.3 - 20.3i)T^{2} \)
53 \( 1 + (5.59 - 11.6i)T + (-33.0 - 41.4i)T^{2} \)
59 \( 1 - 5.20T + 59T^{2} \)
61 \( 1 + (-10.0 - 8.00i)T + (13.5 + 59.4i)T^{2} \)
67 \( 1 + (-3.77 + 0.861i)T + (60.3 - 29.0i)T^{2} \)
71 \( 1 + (1.67 - 7.35i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (-5.36 + 2.58i)T + (45.5 - 57.0i)T^{2} \)
79 \( 1 + (5.30 - 1.21i)T + (71.1 - 34.2i)T^{2} \)
83 \( 1 + (-6.33 + 5.04i)T + (18.4 - 80.9i)T^{2} \)
89 \( 1 + (0.927 - 1.92i)T + (-55.4 - 69.5i)T^{2} \)
97 \( 1 + (4.29 + 5.38i)T + (-21.5 + 94.5i)T^{2} \)
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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

−10.54479895858615870324519733944, −9.928254402348183886966499422396, −9.166676144317142140681897074868, −8.388984718252708614093705487973, −6.86962101580665399421479796690, −5.98227116006684948391801677704, −5.02976191744126659209405849209, −4.22403480357816820221973932750, −3.01882549184501493407913314446, −0.74189664723697400595851856471, 0.42539821797952237799290173986, 2.00817118518483202844985733608, 3.60091821808514432012910188939, 5.41121729572371448315947784360, 6.03644200890308302881789585436, 6.81581153977261123724400896763, 7.72346066803528594510215256107, 8.192325364391363789631841476729, 9.403528693007366460552295524470, 10.09191374581728793550638728368

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