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Properties

Label 2-725-145.74-c1-0-6
Degree $2$
Conductor $725$
Sign $-0.605 - 0.795i$
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  + (−0.193 + 0.400i)2-s + (0.974 + 0.777i)3-s + (1.12 + 1.40i)4-s + (−0.5 + 0.240i)6-s + (0.279 + 0.222i)7-s + (−1.64 + 0.376i)8-s + (−0.321 − 1.40i)9-s + (−1.09 + 4.81i)11-s + 2.24i·12-s + (−5.51 − 1.25i)13-s + (−0.143 + 0.0689i)14-s + (−0.634 + 2.77i)16-s + 4.49i·17-s + (0.626 + 0.143i)18-s + (1.46 + 1.84i)19-s + ⋯
L(s)  = 1  + (−0.136 + 0.283i)2-s + (0.562 + 0.448i)3-s + (0.561 + 0.704i)4-s + (−0.204 + 0.0983i)6-s + (0.105 + 0.0841i)7-s + (−0.583 + 0.133i)8-s + (−0.107 − 0.469i)9-s + (−0.331 + 1.45i)11-s + 0.648i·12-s + (−1.52 − 0.348i)13-s + (−0.0382 + 0.0184i)14-s + (−0.158 + 0.694i)16-s + 1.08i·17-s + (0.147 + 0.0337i)18-s + (0.337 + 0.422i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(725\)    =    \(5^{2} \cdot 29\)
Sign: $-0.605 - 0.795i$
Analytic conductor: \(5.78915\)
Root analytic conductor: \(2.40606\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{725} (74, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 725,\ (\ :1/2),\ -0.605 - 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.721919 + 1.45593i\)
\(L(\frac12)\) \(\approx\) \(0.721919 + 1.45593i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-5.09 - 1.73i)T \)
good2 \( 1 + (0.193 - 0.400i)T + (-1.24 - 1.56i)T^{2} \)
3 \( 1 + (-0.974 - 0.777i)T + (0.667 + 2.92i)T^{2} \)
7 \( 1 + (-0.279 - 0.222i)T + (1.55 + 6.82i)T^{2} \)
11 \( 1 + (1.09 - 4.81i)T + (-9.91 - 4.77i)T^{2} \)
13 \( 1 + (5.51 + 1.25i)T + (11.7 + 5.64i)T^{2} \)
17 \( 1 - 4.49iT - 17T^{2} \)
19 \( 1 + (-1.46 - 1.84i)T + (-4.22 + 18.5i)T^{2} \)
23 \( 1 + (-0.996 - 2.06i)T + (-14.3 + 17.9i)T^{2} \)
31 \( 1 + (-6.02 - 2.90i)T + (19.3 + 24.2i)T^{2} \)
37 \( 1 + (4.81 - 1.09i)T + (33.3 - 16.0i)T^{2} \)
41 \( 1 - 3.10T + 41T^{2} \)
43 \( 1 + (1.47 + 3.06i)T + (-26.8 + 33.6i)T^{2} \)
47 \( 1 + (-6.28 - 1.43i)T + (42.3 + 20.3i)T^{2} \)
53 \( 1 + (-2.03 + 4.22i)T + (-33.0 - 41.4i)T^{2} \)
59 \( 1 - 12.4T + 59T^{2} \)
61 \( 1 + (1.02 - 1.28i)T + (-13.5 - 59.4i)T^{2} \)
67 \( 1 + (2.26 - 0.516i)T + (60.3 - 29.0i)T^{2} \)
71 \( 1 + (-1.63 + 7.15i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (-2.44 - 5.06i)T + (-45.5 + 57.0i)T^{2} \)
79 \( 1 + (1.03 + 4.54i)T + (-71.1 + 34.2i)T^{2} \)
83 \( 1 + (-3.48 + 2.77i)T + (18.4 - 80.9i)T^{2} \)
89 \( 1 + (5.11 + 2.46i)T + (55.4 + 69.5i)T^{2} \)
97 \( 1 + (-0.141 + 0.112i)T + (21.5 - 94.5i)T^{2} \)
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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

−10.31621005614001157926562800968, −9.940916598652022810939619759310, −8.868574255685305721107466340387, −8.108501384379114044936386076673, −7.30559395048894039020201760496, −6.58103762433894092476339826920, −5.24125696051356561214492325004, −4.16819607337379501871998424195, −3.09137976285492812063000095922, −2.14853453383160706117915099986, 0.78718340929108360264941588753, 2.45674520430436713929987062051, 2.85278400798487624887898166388, 4.76094213064641391626151570591, 5.58645363049075259772156805608, 6.74113066727470708794196679922, 7.46674491160597789653225406309, 8.387845849920446620938204225308, 9.302220549919939854776104753532, 10.12430949097290909382518691860

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