Properties

Label 2-925-37.3-c1-0-32
Degree $2$
Conductor $925$
Sign $0.724 + 0.689i$
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  + (1.50 − 1.79i)2-s + (−0.670 + 0.562i)3-s + (−0.610 − 3.46i)4-s + 2.05i·6-s + (4.17 + 1.51i)7-s + (−3.08 − 1.77i)8-s + (−0.387 + 2.19i)9-s + (−2.58 + 4.47i)11-s + (2.35 + 1.97i)12-s + (4.66 − 0.823i)13-s + (9.03 − 5.21i)14-s + (−1.24 + 0.453i)16-s + (1.88 + 0.331i)17-s + (3.37 + 4.01i)18-s + (1.95 + 2.33i)19-s + ⋯
L(s)  = 1  + (1.06 − 1.27i)2-s + (−0.387 + 0.324i)3-s + (−0.305 − 1.73i)4-s + 0.839i·6-s + (1.57 + 0.574i)7-s + (−1.08 − 0.628i)8-s + (−0.129 + 0.733i)9-s + (−0.779 + 1.34i)11-s + (0.680 + 0.571i)12-s + (1.29 − 0.228i)13-s + (2.41 − 1.39i)14-s + (−0.311 + 0.113i)16-s + (0.456 + 0.0804i)17-s + (0.794 + 0.947i)18-s + (0.448 + 0.534i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.724 + 0.689i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.724 + 0.689i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(925\)    =    \(5^{2} \cdot 37\)
Sign: $0.724 + 0.689i$
Analytic conductor: \(7.38616\)
Root analytic conductor: \(2.71774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{925} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 925,\ (\ :1/2),\ 0.724 + 0.689i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.61615 - 1.04623i\)
\(L(\frac12)\) \(\approx\) \(2.61615 - 1.04623i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.05 + 4.53i)T \)
good2 \( 1 + (-1.50 + 1.79i)T + (-0.347 - 1.96i)T^{2} \)
3 \( 1 + (0.670 - 0.562i)T + (0.520 - 2.95i)T^{2} \)
7 \( 1 + (-4.17 - 1.51i)T + (5.36 + 4.49i)T^{2} \)
11 \( 1 + (2.58 - 4.47i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-4.66 + 0.823i)T + (12.2 - 4.44i)T^{2} \)
17 \( 1 + (-1.88 - 0.331i)T + (15.9 + 5.81i)T^{2} \)
19 \( 1 + (-1.95 - 2.33i)T + (-3.29 + 18.7i)T^{2} \)
23 \( 1 + (8.02 - 4.63i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.53 + 1.46i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 5.61iT - 31T^{2} \)
41 \( 1 + (-0.132 - 0.753i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + 0.369iT - 43T^{2} \)
47 \( 1 + (5.66 + 9.80i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.35 + 1.58i)T + (40.6 - 34.0i)T^{2} \)
59 \( 1 + (1.52 + 4.19i)T + (-45.1 + 37.9i)T^{2} \)
61 \( 1 + (2.71 - 0.478i)T + (57.3 - 20.8i)T^{2} \)
67 \( 1 + (-4.24 - 1.54i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (3.35 - 2.81i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 - 2.68T + 73T^{2} \)
79 \( 1 + (-0.687 + 1.88i)T + (-60.5 - 50.7i)T^{2} \)
83 \( 1 + (0.321 - 1.82i)T + (-77.9 - 28.3i)T^{2} \)
89 \( 1 + (-2.11 - 5.81i)T + (-68.1 + 57.2i)T^{2} \)
97 \( 1 + (2.46 - 1.42i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.29851106060339617247675154722, −9.666509449668653456177788593947, −8.104672967765203267561492436851, −7.75897175109862541429698725837, −5.69718201276642428452559342233, −5.46446404129904364802368163875, −4.53516295372186785759721321646, −3.79832451333117132791537668576, −2.23672173830902865403804560920, −1.70602744300241942915191629390, 1.15573870228204325251114974270, 3.26241886797789995200958227756, 4.18974232049740493589325182257, 5.11759505863893729836704045126, 5.91147325890181804931825194102, 6.46373723995171472119768434812, 7.57902033798593396624735422204, 8.144026759985555921697543443639, 8.788103194797434038499667285214, 10.45755822856397969283736537916

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