Properties

Label 2-375-25.16-c1-0-7
Degree $2$
Conductor $375$
Sign $-0.0815 - 0.996i$
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  + (1.74 + 1.26i)2-s + (0.309 + 0.951i)3-s + (0.824 + 2.53i)4-s + (−0.667 + 2.05i)6-s + 3.16·7-s + (−0.446 + 1.37i)8-s + (−0.809 + 0.587i)9-s + (−1.24 − 0.904i)11-s + (−2.15 + 1.56i)12-s + (−4.24 + 3.08i)13-s + (5.52 + 4.01i)14-s + (1.79 − 1.30i)16-s + (−0.398 + 1.22i)17-s − 2.16·18-s + (1.68 − 5.17i)19-s + ⋯
L(s)  = 1  + (1.23 + 0.897i)2-s + (0.178 + 0.549i)3-s + (0.412 + 1.26i)4-s + (−0.272 + 0.838i)6-s + 1.19·7-s + (−0.157 + 0.485i)8-s + (−0.269 + 0.195i)9-s + (−0.375 − 0.272i)11-s + (−0.623 + 0.452i)12-s + (−1.17 + 0.854i)13-s + (1.47 + 1.07i)14-s + (0.448 − 0.325i)16-s + (−0.0967 + 0.297i)17-s − 0.509·18-s + (0.386 − 1.18i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0815 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0815 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(375\)    =    \(3 \cdot 5^{3}\)
Sign: $-0.0815 - 0.996i$
Analytic conductor: \(2.99439\)
Root analytic conductor: \(1.73043\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{375} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 375,\ (\ :1/2),\ -0.0815 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.87803 + 2.03790i\)
\(L(\frac12)\) \(\approx\) \(1.87803 + 2.03790i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

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

−11.80271404617285052386038259244, −10.88470178493192953187043306431, −9.770116898384523279482606122484, −8.589937621179883600833141828688, −7.65537688618624624216675178256, −6.77864819481737098724745757308, −5.48952062098236672792841965071, −4.78381364781322970629013625555, −4.05385992697590758206549441944, −2.47056822529885328366454577796, 1.68647252139506365619481169771, 2.69339067349197327009847472792, 4.03038692839700994861919725122, 5.13019456215415952668293503695, 5.80851397734466664114726608884, 7.59761289158894048615759740986, 7.970281723956818520789943396216, 9.602200608339763406394980103799, 10.60188250317366830754657605372, 11.41728446081458828913855504413

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