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Properties

Label 2-725-29.25-c1-0-26
Degree $2$
Conductor $725$
Sign $0.692 + 0.721i$
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  + (0.178 − 0.781i)2-s + (−1.22 + 0.588i)3-s + (1.22 + 0.588i)4-s + (0.242 + 1.06i)6-s + (4.27 − 2.05i)7-s + (1.67 − 2.10i)8-s + (−0.722 + 0.906i)9-s + (−0.568 − 0.712i)11-s − 1.84·12-s + (−3.71 − 4.66i)13-s + (−0.846 − 3.70i)14-s + (0.346 + 0.433i)16-s + 5.74·17-s + (0.579 + 0.726i)18-s + (−1.98 − 0.953i)19-s + ⋯
L(s)  = 1  + (0.126 − 0.552i)2-s + (−0.705 + 0.339i)3-s + (0.611 + 0.294i)4-s + (0.0988 + 0.433i)6-s + (1.61 − 0.777i)7-s + (0.593 − 0.744i)8-s + (−0.240 + 0.302i)9-s + (−0.171 − 0.214i)11-s − 0.531·12-s + (−1.03 − 1.29i)13-s + (−0.226 − 0.990i)14-s + (0.0865 + 0.108i)16-s + 1.39·17-s + (0.136 + 0.171i)18-s + (−0.454 − 0.218i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.692 + 0.721i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.692 + 0.721i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(725\)    =    \(5^{2} \cdot 29\)
Sign: $0.692 + 0.721i$
Analytic conductor: \(5.78915\)
Root analytic conductor: \(2.40606\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{725} (576, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 725,\ (\ :1/2),\ 0.692 + 0.721i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.62710 - 0.693625i\)
\(L(\frac12)\) \(\approx\) \(1.62710 - 0.693625i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-5.36 + 0.472i)T \)
good2 \( 1 + (-0.178 + 0.781i)T + (-1.80 - 0.867i)T^{2} \)
3 \( 1 + (1.22 - 0.588i)T + (1.87 - 2.34i)T^{2} \)
7 \( 1 + (-4.27 + 2.05i)T + (4.36 - 5.47i)T^{2} \)
11 \( 1 + (0.568 + 0.712i)T + (-2.44 + 10.7i)T^{2} \)
13 \( 1 + (3.71 + 4.66i)T + (-2.89 + 12.6i)T^{2} \)
17 \( 1 - 5.74T + 17T^{2} \)
19 \( 1 + (1.98 + 0.953i)T + (11.8 + 14.8i)T^{2} \)
23 \( 1 + (-1.01 - 4.46i)T + (-20.7 + 9.97i)T^{2} \)
31 \( 1 + (0.548 - 2.40i)T + (-27.9 - 13.4i)T^{2} \)
37 \( 1 + (4.94 - 6.20i)T + (-8.23 - 36.0i)T^{2} \)
41 \( 1 - 5.31T + 41T^{2} \)
43 \( 1 + (2.06 + 9.05i)T + (-38.7 + 18.6i)T^{2} \)
47 \( 1 + (5.01 + 6.29i)T + (-10.4 + 45.8i)T^{2} \)
53 \( 1 + (1.39 - 6.11i)T + (-47.7 - 22.9i)T^{2} \)
59 \( 1 - 4.84T + 59T^{2} \)
61 \( 1 + (-3.79 + 1.82i)T + (38.0 - 47.6i)T^{2} \)
67 \( 1 + (-3.67 + 4.61i)T + (-14.9 - 65.3i)T^{2} \)
71 \( 1 + (-6.91 - 8.66i)T + (-15.7 + 69.2i)T^{2} \)
73 \( 1 + (0.519 + 2.27i)T + (-65.7 + 31.6i)T^{2} \)
79 \( 1 + (5.66 - 7.10i)T + (-17.5 - 77.0i)T^{2} \)
83 \( 1 + (-7.88 - 3.79i)T + (51.7 + 64.8i)T^{2} \)
89 \( 1 + (2 - 8.76i)T + (-80.1 - 38.6i)T^{2} \)
97 \( 1 + (9.39 + 4.52i)T + (60.4 + 75.8i)T^{2} \)
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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

−10.45594731311131161626643330359, −10.06841111731697258937206219875, −8.190234144732402395638843488524, −7.84107804925315369500410134132, −6.90778182510374417455911818124, −5.38577293520578529010947586815, −5.00051991905728317972174808666, −3.72860070541125045684778475957, −2.51645399179367801925229821467, −1.10263791252270781347559287139, 1.49510837175493453675103800907, 2.51026496137704342276732839683, 4.60423620324792087099880387165, 5.24881038164577029862397814974, 6.05914344918173165364647274922, 6.90965472360427353166484713981, 7.76129185799527367508005849911, 8.527961659739133683301899284999, 9.680066908646051070458761119644, 10.79781449994491192463709635410

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