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Properties

Label 2-925-185.84-c1-0-53
Degree $2$
Conductor $925$
Sign $-0.987 + 0.154i$
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  + (0.193 + 0.111i)2-s + (1.82 − 1.05i)3-s + (−0.974 − 1.68i)4-s + 0.471·6-s + (−1.18 + 0.684i)7-s − 0.883i·8-s + (0.719 − 1.24i)9-s − 6.06·11-s + (−3.55 − 2.05i)12-s + (−2.36 + 1.36i)13-s − 0.306·14-s + (−1.85 + 3.20i)16-s + (−2.74 − 1.58i)17-s + (0.278 − 0.160i)18-s + (−3.83 − 6.64i)19-s + ⋯
L(s)  = 1  + (0.136 + 0.0790i)2-s + (1.05 − 0.608i)3-s + (−0.487 − 0.844i)4-s + 0.192·6-s + (−0.448 + 0.258i)7-s − 0.312i·8-s + (0.239 − 0.415i)9-s − 1.82·11-s + (−1.02 − 0.592i)12-s + (−0.655 + 0.378i)13-s − 0.0818·14-s + (−0.462 + 0.801i)16-s + (−0.666 − 0.384i)17-s + (0.0656 − 0.0379i)18-s + (−0.880 − 1.52i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 + 0.154i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.987 + 0.154i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(925\)    =    \(5^{2} \cdot 37\)
Sign: $-0.987 + 0.154i$
Analytic conductor: \(7.38616\)
Root analytic conductor: \(2.71774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{925} (824, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 925,\ (\ :1/2),\ -0.987 + 0.154i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0686220 - 0.880942i\)
\(L(\frac12)\) \(\approx\) \(0.0686220 - 0.880942i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-5.41 + 2.76i)T \)
good2 \( 1 + (-0.193 - 0.111i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (-1.82 + 1.05i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 + (1.18 - 0.684i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + 6.06T + 11T^{2} \)
13 \( 1 + (2.36 - 1.36i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.74 + 1.58i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.83 + 6.64i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 5.25iT - 23T^{2} \)
29 \( 1 - 8.70T + 29T^{2} \)
31 \( 1 - 2.28T + 31T^{2} \)
41 \( 1 + (-2.78 - 4.82i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 10.6iT - 43T^{2} \)
47 \( 1 + 4.35iT - 47T^{2} \)
53 \( 1 + (6.18 + 3.57i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.55 + 4.42i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.52 - 2.63i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.0815 - 0.0470i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.98 - 3.44i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 9.76iT - 73T^{2} \)
79 \( 1 + (-1.40 - 2.43i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (9.71 + 5.60i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (4.04 - 7.00i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 9.94iT - 97T^{2} \)
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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

−9.550743771505651448904245575088, −8.754391427631163062834483457406, −8.187280512464924234291296619611, −7.08361736700560346121697083570, −6.39463934862711322847819456297, −5.10984313690215461073842323332, −4.51488862994285751891605922546, −2.70586326512633763396918292168, −2.35089630918988129660997916383, −0.31996758937710013751203801450, 2.51689776674415567612606196071, 3.10428242965924488568320363027, 4.09603607594887681398549191341, 4.86445771386168681784686238159, 6.11180287765386059767392913512, 7.53971364763046283623590432233, 8.110134035684896052077437447752, 8.600825716078893562319054317481, 9.819642212601216146717099637988, 10.05709646461150858691182150178

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