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Properties

Label 2-375-25.9-c1-0-2
Degree $2$
Conductor $375$
Sign $-0.890 - 0.454i$
Analytic cond. $2.99439$
Root an. cond. $1.73043$
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  + (−1.26 + 1.74i)2-s + (0.951 − 0.309i)3-s + (−0.824 − 2.53i)4-s + (−0.667 + 2.05i)6-s + 3.16i·7-s + (1.37 + 0.446i)8-s + (0.809 − 0.587i)9-s + (−1.24 − 0.904i)11-s + (−1.56 − 2.15i)12-s + (3.08 + 4.24i)13-s + (−5.52 − 4.01i)14-s + (1.79 − 1.30i)16-s + (−1.22 − 0.398i)17-s + 2.16i·18-s + (−1.68 + 5.17i)19-s + ⋯
L(s)  = 1  + (−0.897 + 1.23i)2-s + (0.549 − 0.178i)3-s + (−0.412 − 1.26i)4-s + (−0.272 + 0.838i)6-s + 1.19i·7-s + (0.485 + 0.157i)8-s + (0.269 − 0.195i)9-s + (−0.375 − 0.272i)11-s + (−0.452 − 0.623i)12-s + (0.854 + 1.17i)13-s + (−1.47 − 1.07i)14-s + (0.448 − 0.325i)16-s + (−0.297 − 0.0967i)17-s + 0.509i·18-s + (−0.386 + 1.18i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.890 - 0.454i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.890 - 0.454i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(375\)    =    \(3 \cdot 5^{3}\)
Sign: $-0.890 - 0.454i$
Analytic conductor: \(2.99439\)
Root analytic conductor: \(1.73043\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{375} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 375,\ (\ :1/2),\ -0.890 - 0.454i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.202435 + 0.841854i\)
\(L(\frac12)\) \(\approx\) \(0.202435 + 0.841854i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.951 + 0.309i)T \)
5 \( 1 \)
good2 \( 1 + (1.26 - 1.74i)T + (-0.618 - 1.90i)T^{2} \)
7 \( 1 - 3.16iT - 7T^{2} \)
11 \( 1 + (1.24 + 0.904i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (-3.08 - 4.24i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (1.22 + 0.398i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (1.68 - 5.17i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (3.78 - 5.21i)T + (-7.10 - 21.8i)T^{2} \)
29 \( 1 + (-0.730 - 2.24i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (1.37 - 4.24i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (3.50 + 4.81i)T + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (-6.90 + 5.01i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 8.48iT - 43T^{2} \)
47 \( 1 + (-0.716 + 0.232i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (9.27 - 3.01i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (-3.32 + 2.41i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-8.65 - 6.28i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (-1.80 - 0.586i)T + (54.2 + 39.3i)T^{2} \)
71 \( 1 + (0.0219 + 0.0674i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-2.35 + 3.24i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (0.500 + 1.53i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-12.5 - 4.06i)T + (67.1 + 48.7i)T^{2} \)
89 \( 1 + (-5.88 - 4.27i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-9.69 + 3.15i)T + (78.4 - 57.0i)T^{2} \)
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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

−11.81435696498878830252987551479, −10.53236504635938537184916077127, −9.351592910189826080500668661880, −8.850214103989031334827213278387, −8.183891676434208304818799664419, −7.20791919286401092769037111045, −6.19712935366790383065985579608, −5.47789994220329354772057535670, −3.67038274288408288167299661554, −1.93588534251104317096257986010, 0.74438772769115360300374545921, 2.37465443856375318529832906021, 3.47437333830142445152389950048, 4.54843635771632007042620408572, 6.36043203321891323965674105888, 7.77983327709667588517090647778, 8.328473627365787585770727085758, 9.392806070692167611666976091119, 10.24267090373491945601890558119, 10.72676859205632885157941073211

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