Properties

Label 2-925-185.43-c1-0-50
Degree $2$
Conductor $925$
Sign $-0.999 + 0.0291i$
Analytic cond. $7.38616$
Root an. cond. $2.71774$
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  − 1.72i·2-s + (1.57 − 1.57i)3-s − 0.968·4-s + (−2.71 − 2.71i)6-s + (−0.0954 + 0.0954i)7-s − 1.77i·8-s − 1.96i·9-s + 1.33i·11-s + (−1.52 + 1.52i)12-s − 5.82i·13-s + (0.164 + 0.164i)14-s − 4.99·16-s + 1.40·17-s − 3.38·18-s + (0.447 − 0.447i)19-s + ⋯
L(s)  = 1  − 1.21i·2-s + (0.909 − 0.909i)3-s − 0.484·4-s + (−1.10 − 1.10i)6-s + (−0.0360 + 0.0360i)7-s − 0.628i·8-s − 0.655i·9-s + 0.402i·11-s + (−0.440 + 0.440i)12-s − 1.61i·13-s + (0.0439 + 0.0439i)14-s − 1.24·16-s + 0.341·17-s − 0.798·18-s + (0.102 − 0.102i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0291i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0291i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(925\)    =    \(5^{2} \cdot 37\)
Sign: $-0.999 + 0.0291i$
Analytic conductor: \(7.38616\)
Root analytic conductor: \(2.71774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{925} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 925,\ (\ :1/2),\ -0.999 + 0.0291i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0314268 - 2.15290i\)
\(L(\frac12)\) \(\approx\) \(0.0314268 - 2.15290i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
37 \( 1 + (1.33 - 5.93i)T \)
good2 \( 1 + 1.72iT - 2T^{2} \)
3 \( 1 + (-1.57 + 1.57i)T - 3iT^{2} \)
7 \( 1 + (0.0954 - 0.0954i)T - 7iT^{2} \)
11 \( 1 - 1.33iT - 11T^{2} \)
13 \( 1 + 5.82iT - 13T^{2} \)
17 \( 1 - 1.40T + 17T^{2} \)
19 \( 1 + (-0.447 + 0.447i)T - 19iT^{2} \)
23 \( 1 + 0.235iT - 23T^{2} \)
29 \( 1 + (2.90 + 2.90i)T + 29iT^{2} \)
31 \( 1 + (-0.313 + 0.313i)T - 31iT^{2} \)
41 \( 1 + 0.651iT - 41T^{2} \)
43 \( 1 + 5.82iT - 43T^{2} \)
47 \( 1 + (1.91 - 1.91i)T - 47iT^{2} \)
53 \( 1 + (-2.77 - 2.77i)T + 53iT^{2} \)
59 \( 1 + (-10.3 + 10.3i)T - 59iT^{2} \)
61 \( 1 + (6.94 - 6.94i)T - 61iT^{2} \)
67 \( 1 + (1.08 + 1.08i)T + 67iT^{2} \)
71 \( 1 - 5.92T + 71T^{2} \)
73 \( 1 + (-4.45 + 4.45i)T - 73iT^{2} \)
79 \( 1 + (-0.750 + 0.750i)T - 79iT^{2} \)
83 \( 1 + (-8.16 - 8.16i)T + 83iT^{2} \)
89 \( 1 + (-10.6 - 10.6i)T + 89iT^{2} \)
97 \( 1 + 4.27T + 97T^{2} \)
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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

−9.838621555315920285135937629138, −8.909994478804913436203116614071, −7.968854670783793564678807744244, −7.39243801669589832478653255685, −6.36428765605248026557745364409, −5.08876768929236713758626956522, −3.67843805437016205756880253424, −2.89163486996704390667126173705, −2.09426385667795164169003969000, −0.929309763235031490286799504188, 2.12439312587705149602139718825, 3.44342846139778861694857378441, 4.35068644362138939901687954053, 5.28946733129992666659849637868, 6.33406119417975299483508743563, 7.10880685494628153288381006449, 8.012908524212641283660370306979, 8.848493445598441310223680071630, 9.244185597679622265454554122543, 10.20276188430496119300165030302

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