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Properties

Label 2-925-185.99-c1-0-55
Degree $2$
Conductor $925$
Sign $-0.962 + 0.270i$
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  + (2.02 − 1.69i)2-s + (1.25 − 1.49i)3-s + (0.865 − 4.90i)4-s − 5.16i·6-s + (−0.868 − 2.38i)7-s + (−3.94 − 6.83i)8-s + (−0.143 − 0.812i)9-s + (1.76 + 3.05i)11-s + (−6.26 − 7.46i)12-s + (−0.828 + 4.70i)13-s + (−5.81 − 3.35i)14-s + (−10.2 − 3.72i)16-s + (0.397 + 2.25i)17-s + (−1.67 − 1.40i)18-s + (4.07 − 4.86i)19-s + ⋯
L(s)  = 1  + (1.43 − 1.20i)2-s + (0.725 − 0.865i)3-s + (0.432 − 2.45i)4-s − 2.11i·6-s + (−0.328 − 0.902i)7-s + (−1.39 − 2.41i)8-s + (−0.0477 − 0.270i)9-s + (0.531 + 0.920i)11-s + (−1.80 − 2.15i)12-s + (−0.229 + 1.30i)13-s + (−1.55 − 0.897i)14-s + (−2.55 − 0.930i)16-s + (0.0963 + 0.546i)17-s + (−0.393 − 0.330i)18-s + (0.935 − 1.11i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.962 + 0.270i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.962 + 0.270i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(925\)    =    \(5^{2} \cdot 37\)
Sign: $-0.962 + 0.270i$
Analytic conductor: \(7.38616\)
Root analytic conductor: \(2.71774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{925} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 925,\ (\ :1/2),\ -0.962 + 0.270i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.572348 - 4.15070i\)
\(L(\frac12)\) \(\approx\) \(0.572348 - 4.15070i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.95 - 4.62i)T \)
good2 \( 1 + (-2.02 + 1.69i)T + (0.347 - 1.96i)T^{2} \)
3 \( 1 + (-1.25 + 1.49i)T + (-0.520 - 2.95i)T^{2} \)
7 \( 1 + (0.868 + 2.38i)T + (-5.36 + 4.49i)T^{2} \)
11 \( 1 + (-1.76 - 3.05i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.828 - 4.70i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (-0.397 - 2.25i)T + (-15.9 + 5.81i)T^{2} \)
19 \( 1 + (-4.07 + 4.86i)T + (-3.29 - 18.7i)T^{2} \)
23 \( 1 + (3.48 - 6.03i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.56 + 1.48i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 7.04iT - 31T^{2} \)
41 \( 1 + (-0.250 + 1.42i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + 4.36T + 43T^{2} \)
47 \( 1 + (8.51 + 4.91i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.27 - 11.7i)T + (-40.6 - 34.0i)T^{2} \)
59 \( 1 + (0.273 - 0.752i)T + (-45.1 - 37.9i)T^{2} \)
61 \( 1 + (6.93 + 1.22i)T + (57.3 + 20.8i)T^{2} \)
67 \( 1 + (2.01 + 5.54i)T + (-51.3 + 43.0i)T^{2} \)
71 \( 1 + (4.93 + 4.14i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 - 3.18iT - 73T^{2} \)
79 \( 1 + (5.14 + 14.1i)T + (-60.5 + 50.7i)T^{2} \)
83 \( 1 + (-9.56 + 1.68i)T + (77.9 - 28.3i)T^{2} \)
89 \( 1 + (4.54 - 12.4i)T + (-68.1 - 57.2i)T^{2} \)
97 \( 1 + (-1.61 + 2.79i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.774558961935205819860317915941, −9.321991756156123212938104435051, −7.72195276762761875095816302054, −6.96021937962565657145153646872, −6.23121145829388110540548841908, −4.82739742715454971505511973735, −4.15108994515801110223071834045, −3.20125797083255242561827789546, −2.10033434208766959150997907317, −1.34942580376548275376711769813, 2.99316822736680018601819388834, 3.20403212892860048122546083871, 4.30825320218640890369095897091, 5.29479135997251664356741959006, 5.93429354872889282466537502865, 6.76531495062443099258736910149, 8.066048349408629223246505013394, 8.405652334671305675891119117495, 9.384064287030530143039812367704, 10.32313036561069276169645077025

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