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Properties

Label 2-375-25.9-c1-0-3
Degree $2$
Conductor $375$
Sign $-0.147 - 0.989i$
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  + (−0.0832 + 0.114i)2-s + (−0.951 + 0.309i)3-s + (0.611 + 1.88i)4-s + (0.0437 − 0.134i)6-s − 0.858i·7-s + (−0.536 − 0.174i)8-s + (0.809 − 0.587i)9-s + (2.97 + 2.16i)11-s + (−1.16 − 1.60i)12-s + (2.69 + 3.70i)13-s + (0.0983 + 0.0714i)14-s + (−3.13 + 2.28i)16-s + (−5.04 − 1.63i)17-s + 0.141i·18-s + (−1.96 + 6.05i)19-s + ⋯
L(s)  = 1  + (−0.0588 + 0.0810i)2-s + (−0.549 + 0.178i)3-s + (0.305 + 0.941i)4-s + (0.0178 − 0.0550i)6-s − 0.324i·7-s + (−0.189 − 0.0616i)8-s + (0.269 − 0.195i)9-s + (0.897 + 0.652i)11-s + (−0.335 − 0.462i)12-s + (0.746 + 1.02i)13-s + (0.0262 + 0.0191i)14-s + (−0.784 + 0.570i)16-s + (−1.22 − 0.397i)17-s + 0.0333i·18-s + (−0.451 + 1.38i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.147 - 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.147 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(375\)    =    \(3 \cdot 5^{3}\)
Sign: $-0.147 - 0.989i$
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: \((2,\ 375,\ (\ :1/2),\ -0.147 - 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.720090 + 0.835326i\)
\(L(\frac12)\) \(\approx\) \(0.720090 + 0.835326i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.0832 - 0.114i)T + (-0.618 - 1.90i)T^{2} \)
7 \( 1 + 0.858iT - 7T^{2} \)
11 \( 1 + (-2.97 - 2.16i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (-2.69 - 3.70i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (5.04 + 1.63i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (1.96 - 6.05i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (2.01 - 2.76i)T + (-7.10 - 21.8i)T^{2} \)
29 \( 1 + (1.15 + 3.55i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-0.387 + 1.19i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-4.37 - 6.02i)T + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (-2.04 + 1.48i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 3.37iT - 43T^{2} \)
47 \( 1 + (-8.08 + 2.62i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (2.23 - 0.725i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (-10.6 + 7.71i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (8.37 + 6.08i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (-3.18 - 1.03i)T + (54.2 + 39.3i)T^{2} \)
71 \( 1 + (1.33 + 4.11i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-5.33 + 7.34i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (-1.00 - 3.08i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-7.03 - 2.28i)T + (67.1 + 48.7i)T^{2} \)
89 \( 1 + (12.5 + 9.10i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-9.54 + 3.10i)T + (78.4 - 57.0i)T^{2} \)
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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

−11.64766601853260432579393318678, −10.97135747955781234225660795142, −9.734054099116724308343799719833, −8.891638773829089884975077172031, −7.81123567925127155864608777946, −6.78721679612577057246531653518, −6.16873474538322490337417917868, −4.37734571291075028874650627830, −3.82815377173622186846835941475, −1.94280719294863780903306111270, 0.836576994768964584838111765476, 2.44864817358219722792783186956, 4.22474589284006673434555124122, 5.52947795979585500932415898203, 6.21992437136362636429639622157, 7.02171107942744795481488166885, 8.593691512893230239021583283377, 9.227641477799377615881017776683, 10.65565058551286596608334614387, 10.90381628702732445217725101509

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