Properties

Label 2-925-37.3-c1-0-45
Degree $2$
Conductor $925$
Sign $0.529 + 0.848i$
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  + (0.301 − 0.359i)2-s + (2.39 − 2.00i)3-s + (0.309 + 1.75i)4-s − 1.46i·6-s + (−1.06 − 0.387i)7-s + (1.53 + 0.886i)8-s + (1.17 − 6.64i)9-s + (0.690 − 1.19i)11-s + (4.25 + 3.57i)12-s + (6.56 − 1.15i)13-s + (−0.460 + 0.265i)14-s + (−2.56 + 0.933i)16-s + (−3.42 − 0.603i)17-s + (−2.03 − 2.42i)18-s + (3.70 + 4.40i)19-s + ⋯
L(s)  = 1  + (0.213 − 0.254i)2-s + (1.38 − 1.15i)3-s + (0.154 + 0.876i)4-s − 0.597i·6-s + (−0.402 − 0.146i)7-s + (0.542 + 0.313i)8-s + (0.390 − 2.21i)9-s + (0.208 − 0.360i)11-s + (1.22 + 1.03i)12-s + (1.82 − 0.321i)13-s + (−0.122 + 0.0710i)14-s + (−0.640 + 0.233i)16-s + (−0.830 − 0.146i)17-s + (−0.479 − 0.571i)18-s + (0.848 + 1.01i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.529 + 0.848i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.529 + 0.848i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(925\)    =    \(5^{2} \cdot 37\)
Sign: $0.529 + 0.848i$
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: \((2,\ 925,\ (\ :1/2),\ 0.529 + 0.848i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.60869 - 1.44606i\)
\(L(\frac12)\) \(\approx\) \(2.60869 - 1.44606i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-5.05 + 3.37i)T \)
good2 \( 1 + (-0.301 + 0.359i)T + (-0.347 - 1.96i)T^{2} \)
3 \( 1 + (-2.39 + 2.00i)T + (0.520 - 2.95i)T^{2} \)
7 \( 1 + (1.06 + 0.387i)T + (5.36 + 4.49i)T^{2} \)
11 \( 1 + (-0.690 + 1.19i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-6.56 + 1.15i)T + (12.2 - 4.44i)T^{2} \)
17 \( 1 + (3.42 + 0.603i)T + (15.9 + 5.81i)T^{2} \)
19 \( 1 + (-3.70 - 4.40i)T + (-3.29 + 18.7i)T^{2} \)
23 \( 1 + (2.26 - 1.30i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (6.17 + 3.56i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 2.69iT - 31T^{2} \)
41 \( 1 + (-1.38 - 7.83i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + 10.8iT - 43T^{2} \)
47 \( 1 + (0.933 + 1.61i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.92 - 0.700i)T + (40.6 - 34.0i)T^{2} \)
59 \( 1 + (-2.08 - 5.74i)T + (-45.1 + 37.9i)T^{2} \)
61 \( 1 + (-6.18 + 1.09i)T + (57.3 - 20.8i)T^{2} \)
67 \( 1 + (5.97 + 2.17i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (3.10 - 2.60i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 - 0.109T + 73T^{2} \)
79 \( 1 + (0.920 - 2.52i)T + (-60.5 - 50.7i)T^{2} \)
83 \( 1 + (2.40 - 13.6i)T + (-77.9 - 28.3i)T^{2} \)
89 \( 1 + (-4.36 - 12.0i)T + (-68.1 + 57.2i)T^{2} \)
97 \( 1 + (3.43 - 1.98i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.663985446995326246335198284035, −8.777240740025276819999688612592, −8.241605133991370250758205835152, −7.58983294507412765731214370091, −6.75103446153192918803518963275, −5.90351254063501693895997360349, −3.81635356721114165023914832804, −3.56811483295045465738011670356, −2.47101215617800619502248080033, −1.37108970756260788575249436407, 1.72063727006746852534563563296, 2.96098914892587067823997409774, 3.99937672894859230481575331018, 4.67323229706290445126126699194, 5.82013762662766969887143342137, 6.73573167576336392481681064203, 7.83339577873134939784826046592, 8.900409885374212658789171760974, 9.281154456017281607650920664073, 9.977287694089208838451650500369

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