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Properties

Label 2-925-185.8-c1-0-41
Degree $2$
Conductor $925$
Sign $-0.890 + 0.455i$
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  + (0.926 − 0.534i)2-s + (−0.561 − 2.09i)3-s + (−0.427 + 0.740i)4-s + (−1.64 − 1.64i)6-s + (0.642 + 2.39i)7-s + 3.05i·8-s + (−1.47 + 0.851i)9-s − 5.02i·11-s + (1.79 + 0.480i)12-s + (−3.98 − 2.30i)13-s + (1.87 + 1.87i)14-s + (0.779 + 1.34i)16-s + (−2.61 − 4.53i)17-s + (−0.911 + 1.57i)18-s + (−1.96 − 7.32i)19-s + ⋯
L(s)  = 1  + (0.655 − 0.378i)2-s + (−0.324 − 1.20i)3-s + (−0.213 + 0.370i)4-s + (−0.669 − 0.669i)6-s + (0.242 + 0.905i)7-s + 1.08i·8-s + (−0.491 + 0.283i)9-s − 1.51i·11-s + (0.517 + 0.138i)12-s + (−1.10 − 0.638i)13-s + (0.501 + 0.501i)14-s + (0.194 + 0.337i)16-s + (−0.634 − 1.09i)17-s + (−0.214 + 0.372i)18-s + (−0.450 − 1.68i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.890 + 0.455i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.890 + 0.455i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(925\)    =    \(5^{2} \cdot 37\)
Sign: $-0.890 + 0.455i$
Analytic conductor: \(7.38616\)
Root analytic conductor: \(2.71774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{925} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 925,\ (\ :1/2),\ -0.890 + 0.455i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.299788 - 1.24373i\)
\(L(\frac12)\) \(\approx\) \(0.299788 - 1.24373i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.23 - 4.36i)T \)
good2 \( 1 + (-0.926 + 0.534i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (0.561 + 2.09i)T + (-2.59 + 1.5i)T^{2} \)
7 \( 1 + (-0.642 - 2.39i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + 5.02iT - 11T^{2} \)
13 \( 1 + (3.98 + 2.30i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.61 + 4.53i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.96 + 7.32i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 - 3.90iT - 23T^{2} \)
29 \( 1 + (2.46 + 2.46i)T + 29iT^{2} \)
31 \( 1 + (-6.59 + 6.59i)T - 31iT^{2} \)
41 \( 1 + (-2.12 - 1.22i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 - 2.84iT - 43T^{2} \)
47 \( 1 + (-3.77 + 3.77i)T - 47iT^{2} \)
53 \( 1 + (-0.509 + 1.90i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (12.3 + 3.30i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-3.14 - 11.7i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (-1.37 + 0.369i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-0.476 + 0.824i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (5.74 - 5.74i)T - 73iT^{2} \)
79 \( 1 + (-2.65 - 9.89i)T + (-68.4 + 39.5i)T^{2} \)
83 \( 1 + (-0.433 + 1.61i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + (-0.934 + 3.48i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + 10.8T + 97T^{2} \)
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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

−9.578842836145892863157177535004, −8.721697649670975330934920963719, −7.968427861943154738398768054245, −7.20166959715360582365566454299, −6.10188993265722838270914388796, −5.37166900836272177101857998181, −4.45139388979770105146464719111, −2.84766740209763676870082203601, −2.42912373768458942179170726359, −0.49622384610329626192458589421, 1.80094996438333189932125244132, 3.82419585653326340459978974717, 4.47891392421910819196150161986, 4.74546072713276047040049458872, 5.95090872683708265821146405197, 6.85963977458062351891141599645, 7.68549499444149867985608919268, 9.067098630945479678348149791866, 9.855223905730685960953257927676, 10.37115293963096270130731355491

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