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Properties

Label 2-925-185.64-c1-0-50
Degree $2$
Conductor $925$
Sign $0.205 + 0.978i$
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  + (0.5 − 0.866i)2-s + (2.36 − 1.36i)3-s + (0.500 + 0.866i)4-s − 2.73i·6-s + (3 − 1.73i)7-s + 3·8-s + (2.23 − 3.86i)9-s − 4.73·11-s + (2.36 + 1.36i)12-s + (−1.73 − 3i)13-s − 3.46i·14-s + (0.500 − 0.866i)16-s + (−1.86 + 3.23i)17-s + (−2.23 − 3.86i)18-s + (−1.09 + 0.633i)19-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (1.36 − 0.788i)3-s + (0.250 + 0.433i)4-s − 1.11i·6-s + (1.13 − 0.654i)7-s + 1.06·8-s + (0.744 − 1.28i)9-s − 1.42·11-s + (0.683 + 0.394i)12-s + (−0.480 − 0.832i)13-s − 0.925i·14-s + (0.125 − 0.216i)16-s + (−0.452 + 0.783i)17-s + (−0.526 − 0.911i)18-s + (−0.251 + 0.145i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.205 + 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.205 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(925\)    =    \(5^{2} \cdot 37\)
Sign: $0.205 + 0.978i$
Analytic conductor: \(7.38616\)
Root analytic conductor: \(2.71774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{925} (249, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 925,\ (\ :1/2),\ 0.205 + 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.66772 - 2.16555i\)
\(L(\frac12)\) \(\approx\) \(2.66772 - 2.16555i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-6.06 - 0.5i)T \)
good2 \( 1 + (-0.5 + 0.866i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-2.36 + 1.36i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 + (-3 + 1.73i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + 4.73T + 11T^{2} \)
13 \( 1 + (1.73 + 3i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.86 - 3.23i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.09 - 0.633i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 0.732T + 23T^{2} \)
29 \( 1 - 0.267iT - 29T^{2} \)
31 \( 1 - 8.19iT - 31T^{2} \)
41 \( 1 + (1.5 + 2.59i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 3.46T + 43T^{2} \)
47 \( 1 + 2.19iT - 47T^{2} \)
53 \( 1 + (2.19 + 1.26i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-11.6 - 6.73i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.69 + 2.13i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.29 - 3.63i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (3 + 5.19i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + (3.29 - 1.90i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (7.09 + 4.09i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (7.96 + 4.59i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 4.26T + 97T^{2} \)
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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

−10.21799064045709548849286518372, −8.630507639912809696871256721752, −8.123914076600999895777687894230, −7.61256065795237562586728792625, −6.93406649213794832222489358853, −5.23839744893467107382476997733, −4.24369901903365778842475563656, −3.15818484977355136002681398986, −2.41314229025219460182952698396, −1.48957391401882589639532538526, 2.07782464708775948287400734557, 2.65040485257195168258409490511, 4.33383985282025362958721601617, 4.84165662549991352336859631866, 5.68895717888528306656980294733, 7.04556657699740549266571476966, 7.88450880994395282284414662141, 8.403272864738497118811660063786, 9.436351182834652191519325939172, 9.999707980498789254010455652681

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