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Properties

Label 2-925-185.103-c1-0-33
Degree $2$
Conductor $925$
Sign $0.984 + 0.174i$
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  + (0.567 + 0.983i)2-s + (−0.273 − 1.02i)3-s + (0.354 − 0.614i)4-s + (0.849 − 0.849i)6-s + (0.422 + 1.57i)7-s + 3.07·8-s + (1.63 − 0.941i)9-s − 1.92i·11-s + (−0.724 − 0.194i)12-s + (−0.763 + 1.32i)13-s + (−1.31 + 1.31i)14-s + (1.03 + 1.79i)16-s + (−1.14 + 0.660i)17-s + (1.85 + 1.06i)18-s + (2.17 − 0.583i)19-s + ⋯
L(s)  = 1  + (0.401 + 0.695i)2-s + (−0.157 − 0.589i)3-s + (0.177 − 0.307i)4-s + (0.346 − 0.346i)6-s + (0.159 + 0.596i)7-s + 1.08·8-s + (0.543 − 0.313i)9-s − 0.579i·11-s + (−0.209 − 0.0560i)12-s + (−0.211 + 0.366i)13-s + (−0.350 + 0.350i)14-s + (0.259 + 0.449i)16-s + (−0.277 + 0.160i)17-s + (0.436 + 0.251i)18-s + (0.499 − 0.133i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.174i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 + 0.174i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(925\)    =    \(5^{2} \cdot 37\)
Sign: $0.984 + 0.174i$
Analytic conductor: \(7.38616\)
Root analytic conductor: \(2.71774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{925} (843, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 925,\ (\ :1/2),\ 0.984 + 0.174i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.25395 - 0.197907i\)
\(L(\frac12)\) \(\approx\) \(2.25395 - 0.197907i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.71 + 5.83i)T \)
good2 \( 1 + (-0.567 - 0.983i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (0.273 + 1.02i)T + (-2.59 + 1.5i)T^{2} \)
7 \( 1 + (-0.422 - 1.57i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + 1.92iT - 11T^{2} \)
13 \( 1 + (0.763 - 1.32i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.14 - 0.660i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.17 + 0.583i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 - 4.57T + 23T^{2} \)
29 \( 1 + (-0.241 + 0.241i)T - 29iT^{2} \)
31 \( 1 + (1.00 + 1.00i)T + 31iT^{2} \)
41 \( 1 + (-6.76 - 3.90i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + 5.46T + 43T^{2} \)
47 \( 1 + (1.79 - 1.79i)T - 47iT^{2} \)
53 \( 1 + (-0.870 + 3.24i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (1.40 - 5.26i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-10.4 + 2.80i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (-9.45 + 2.53i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-1.65 + 2.86i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (1.78 - 1.78i)T - 73iT^{2} \)
79 \( 1 + (12.3 - 3.30i)T + (68.4 - 39.5i)T^{2} \)
83 \( 1 + (2.29 - 8.57i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + (16.4 + 4.40i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 - 15.4iT - 97T^{2} \)
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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

−9.993528345283054407811218234637, −9.160391996277057736187628691423, −8.140199910774533742699992159260, −7.21037111709306340345067712633, −6.66036154036930813457319773131, −5.80216973391456581191260445382, −5.06329962031263784898955872973, −3.94206644099852454308025467809, −2.37022127154871676472205704385, −1.14552749406102607190523085503, 1.46800236244811342437560658637, 2.76650006691750133423625491489, 3.83570205923874040604848447743, 4.57071825322813606208802352219, 5.31427158269445395236243700661, 6.97423266162131548161617624705, 7.38636887664130452220665390830, 8.417185832226024280781974787350, 9.661746725231429542278612475948, 10.25013191332102709959285708493

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