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Properties

Label 2-725-145.34-c1-0-22
Degree $2$
Conductor $725$
Sign $0.609 - 0.792i$
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  + (1.21 + 1.52i)2-s + (−0.138 − 0.605i)3-s + (−0.403 + 1.76i)4-s + (0.755 − 0.947i)6-s + (0.123 − 0.0281i)7-s + (0.328 − 0.157i)8-s + (2.35 − 1.13i)9-s + (−0.151 + 0.315i)11-s + 1.12·12-s + (0.178 − 0.369i)13-s + (0.193 + 0.153i)14-s + (3.90 + 1.88i)16-s − 0.332·17-s + (4.60 + 2.21i)18-s + (2.32 + 0.530i)19-s + ⋯
L(s)  = 1  + (0.860 + 1.07i)2-s + (−0.0797 − 0.349i)3-s + (−0.201 + 0.884i)4-s + (0.308 − 0.387i)6-s + (0.0465 − 0.0106i)7-s + (0.115 − 0.0558i)8-s + (0.785 − 0.378i)9-s + (−0.0458 + 0.0951i)11-s + 0.325·12-s + (0.0494 − 0.102i)13-s + (0.0515 + 0.0411i)14-s + (0.977 + 0.470i)16-s − 0.0805·17-s + (1.08 + 0.522i)18-s + (0.533 + 0.121i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.609 - 0.792i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.609 - 0.792i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(725\)    =    \(5^{2} \cdot 29\)
Sign: $0.609 - 0.792i$
Analytic conductor: \(5.78915\)
Root analytic conductor: \(2.40606\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{725} (324, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 725,\ (\ :1/2),\ 0.609 - 0.792i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.32979 + 1.14743i\)
\(L(\frac12)\) \(\approx\) \(2.32979 + 1.14743i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.82 - 3.79i)T \)
good2 \( 1 + (-1.21 - 1.52i)T + (-0.445 + 1.94i)T^{2} \)
3 \( 1 + (0.138 + 0.605i)T + (-2.70 + 1.30i)T^{2} \)
7 \( 1 + (-0.123 + 0.0281i)T + (6.30 - 3.03i)T^{2} \)
11 \( 1 + (0.151 - 0.315i)T + (-6.85 - 8.60i)T^{2} \)
13 \( 1 + (-0.178 + 0.369i)T + (-8.10 - 10.1i)T^{2} \)
17 \( 1 + 0.332T + 17T^{2} \)
19 \( 1 + (-2.32 - 0.530i)T + (17.1 + 8.24i)T^{2} \)
23 \( 1 + (-1.73 - 1.38i)T + (5.11 + 22.4i)T^{2} \)
31 \( 1 + (-4.32 + 3.44i)T + (6.89 - 30.2i)T^{2} \)
37 \( 1 + (6.15 - 2.96i)T + (23.0 - 28.9i)T^{2} \)
41 \( 1 + 5.39iT - 41T^{2} \)
43 \( 1 + (3.17 - 3.97i)T + (-9.56 - 41.9i)T^{2} \)
47 \( 1 + (2.93 + 1.41i)T + (29.3 + 36.7i)T^{2} \)
53 \( 1 + (5.19 - 4.14i)T + (11.7 - 51.6i)T^{2} \)
59 \( 1 - 1.95T + 59T^{2} \)
61 \( 1 + (-13.1 + 3.00i)T + (54.9 - 26.4i)T^{2} \)
67 \( 1 + (2.39 + 4.98i)T + (-41.7 + 52.3i)T^{2} \)
71 \( 1 + (12.6 + 6.11i)T + (44.2 + 55.5i)T^{2} \)
73 \( 1 + (6.13 - 7.69i)T + (-16.2 - 71.1i)T^{2} \)
79 \( 1 + (3.59 + 7.46i)T + (-49.2 + 61.7i)T^{2} \)
83 \( 1 + (-3.14 - 0.716i)T + (74.7 + 36.0i)T^{2} \)
89 \( 1 + (8.98 - 7.16i)T + (19.8 - 86.7i)T^{2} \)
97 \( 1 + (0.0699 - 0.306i)T + (-87.3 - 42.0i)T^{2} \)
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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

−10.42801072958311754713011812532, −9.693046919903664408387048434017, −8.471486167291354632861258892315, −7.56824205274761832681756894813, −6.89142108685527949673091861455, −6.20097877412592101586131064986, −5.18508401352323342540595326518, −4.38107518007203062462889265401, −3.27655537664585064189226415825, −1.40960844506828812705902478693, 1.44128103566492495835868647031, 2.69588902747992608469320396084, 3.73400403423541274133080531540, 4.64581448899027125190550593036, 5.27969233125361249701592461109, 6.63707567787944658478103670053, 7.67296084453568617715271856040, 8.694802720057962538252624160868, 9.996197073656708732223242242531, 10.26902973659277108593748778941

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