Properties

Label 2-925-185.43-c1-0-48
Degree $2$
Conductor $925$
Sign $-0.461 + 0.887i$
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  − i·2-s + (1 − i)3-s + 4-s + (−1 − i)6-s + (3 − 3i)7-s − 3i·8-s + i·9-s − 2i·11-s + (1 − i)12-s − 2i·13-s + (−3 − 3i)14-s − 16-s − 4·17-s + 18-s + (−3 + 3i)19-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.577 − 0.577i)3-s + 0.5·4-s + (−0.408 − 0.408i)6-s + (1.13 − 1.13i)7-s − 1.06i·8-s + 0.333i·9-s − 0.603i·11-s + (0.288 − 0.288i)12-s − 0.554i·13-s + (−0.801 − 0.801i)14-s − 0.250·16-s − 0.970·17-s + 0.235·18-s + (−0.688 + 0.688i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.461 + 0.887i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.461 + 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(925\)    =    \(5^{2} \cdot 37\)
Sign: $-0.461 + 0.887i$
Analytic conductor: \(7.38616\)
Root analytic conductor: \(2.71774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{925} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 925,\ (\ :1/2),\ -0.461 + 0.887i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.28721 - 2.12026i\)
\(L(\frac12)\) \(\approx\) \(1.28721 - 2.12026i\)
\(L(\frac{3}{2})\) 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 \( 1 + (6 + i)T \)
good2 \( 1 + iT - 2T^{2} \)
3 \( 1 + (-1 + i)T - 3iT^{2} \)
7 \( 1 + (-3 + 3i)T - 7iT^{2} \)
11 \( 1 + 2iT - 11T^{2} \)
13 \( 1 + 2iT - 13T^{2} \)
17 \( 1 + 4T + 17T^{2} \)
19 \( 1 + (3 - 3i)T - 19iT^{2} \)
23 \( 1 - 8iT - 23T^{2} \)
29 \( 1 + (-7 - 7i)T + 29iT^{2} \)
31 \( 1 + (-3 + 3i)T - 31iT^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 12iT - 43T^{2} \)
47 \( 1 + (5 - 5i)T - 47iT^{2} \)
53 \( 1 + (-3 - 3i)T + 53iT^{2} \)
59 \( 1 + (7 - 7i)T - 59iT^{2} \)
61 \( 1 + (-1 + i)T - 61iT^{2} \)
67 \( 1 + (-3 - 3i)T + 67iT^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 + (-1 + i)T - 73iT^{2} \)
79 \( 1 + (3 - 3i)T - 79iT^{2} \)
83 \( 1 + (-5 - 5i)T + 83iT^{2} \)
89 \( 1 + (5 + 5i)T + 89iT^{2} \)
97 \( 1 - 8T + 97T^{2} \)
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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

−10.20249859752394532342813727684, −8.812116329072003620362678286001, −8.046445727397075377633807426006, −7.39074223735103369416879323235, −6.65911213018938097791051897353, −5.32182023066900790585127695292, −4.14400858554233655464090153641, −3.17135453686488283845065423486, −1.99958803053858330928332945675, −1.18948019156111807542453953115, 2.06328796973274289544140808536, 2.71879321911847734031924670457, 4.49344072393122078262140021508, 4.89201709379766041711649508464, 6.37058785143761647523147099932, 6.67827569439679165354248158536, 8.174229520824179048202856934599, 8.456562380575418703972603460092, 9.217992416792939184605974473920, 10.29148278287888484959222900612

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