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Properties

Label 2-925-185.174-c1-0-21
Degree $2$
Conductor $925$
Sign $0.733 - 0.679i$
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  + (−0.348 + 0.201i)2-s + (0.227 + 0.131i)3-s + (−0.919 + 1.59i)4-s − 0.105·6-s + (1.99 + 1.15i)7-s − 1.54i·8-s + (−1.46 − 2.53i)9-s + 4.73·11-s + (−0.417 + 0.241i)12-s + (0.756 + 0.436i)13-s − 0.927·14-s + (−1.52 − 2.64i)16-s + (3.04 − 1.75i)17-s + (1.02 + 0.589i)18-s + (1.46 − 2.54i)19-s + ⋯
L(s)  = 1  + (−0.246 + 0.142i)2-s + (0.131 + 0.0757i)3-s + (−0.459 + 0.795i)4-s − 0.0430·6-s + (0.754 + 0.435i)7-s − 0.545i·8-s + (−0.488 − 0.846i)9-s + 1.42·11-s + (−0.120 + 0.0695i)12-s + (0.209 + 0.121i)13-s − 0.247·14-s + (−0.381 − 0.661i)16-s + (0.737 − 0.425i)17-s + (0.240 + 0.138i)18-s + (0.337 − 0.584i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.733 - 0.679i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.733 - 0.679i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(925\)    =    \(5^{2} \cdot 37\)
Sign: $0.733 - 0.679i$
Analytic conductor: \(7.38616\)
Root analytic conductor: \(2.71774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{925} (174, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 925,\ (\ :1/2),\ 0.733 - 0.679i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.39739 + 0.548226i\)
\(L(\frac12)\) \(\approx\) \(1.39739 + 0.548226i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.28 - 5.94i)T \)
good2 \( 1 + (0.348 - 0.201i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (-0.227 - 0.131i)T + (1.5 + 2.59i)T^{2} \)
7 \( 1 + (-1.99 - 1.15i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 - 4.73T + 11T^{2} \)
13 \( 1 + (-0.756 - 0.436i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.04 + 1.75i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.46 + 2.54i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 4.66iT - 23T^{2} \)
29 \( 1 + 1.38T + 29T^{2} \)
31 \( 1 - 4.55T + 31T^{2} \)
41 \( 1 + (-4.21 + 7.29i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 - 8.27iT - 43T^{2} \)
47 \( 1 - 9.58iT - 47T^{2} \)
53 \( 1 + (0.0727 - 0.0419i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.55 - 6.16i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.42 + 4.20i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.47 + 2.58i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (3.82 - 6.62i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 2.22iT - 73T^{2} \)
79 \( 1 + (0.514 - 0.891i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-11.9 + 6.92i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-4.27 - 7.39i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 17.7iT - 97T^{2} \)
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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

−9.736406911426021320232240716364, −9.237363711134401573947944379970, −8.614148308607114342135965662809, −7.79336672562116078470700898024, −6.89003295252150856271312740862, −5.92737960278666164377150501029, −4.75661544888341611605570474715, −3.79963004641416838024194194577, −2.94630904435690386484516312276, −1.14975770068197628934481355011, 1.04778941556795579751446431718, 2.06180290905519156762640089783, 3.74235065092353243348319103281, 4.68928325451976911618853895512, 5.58174381979159719995189227399, 6.44384757031573561677706808926, 7.66600794515909883039308843896, 8.413819120217063091235307693030, 9.080667511360297657057195312201, 10.08903908480139776973088767045

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