Properties

Label 2-375-25.14-c1-0-15
Degree $2$
Conductor $375$
Sign $-0.744 + 0.667i$
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  + (−0.0527 − 0.0726i)2-s + (−0.951 − 0.309i)3-s + (0.615 − 1.89i)4-s + (0.0277 + 0.0854i)6-s − 4.36i·7-s + (−0.341 + 0.110i)8-s + (0.809 + 0.587i)9-s + (−3.55 + 2.58i)11-s + (−1.17 + 1.61i)12-s + (−1.16 + 1.60i)13-s + (−0.316 + 0.230i)14-s + (−3.19 − 2.32i)16-s + (0.948 − 0.308i)17-s − 0.0898i·18-s + (−0.417 − 1.28i)19-s + ⋯
L(s)  = 1  + (−0.0373 − 0.0513i)2-s + (−0.549 − 0.178i)3-s + (0.307 − 0.947i)4-s + (0.0113 + 0.0348i)6-s − 1.64i·7-s + (−0.120 + 0.0391i)8-s + (0.269 + 0.195i)9-s + (−1.07 + 0.778i)11-s + (−0.337 + 0.465i)12-s + (−0.323 + 0.444i)13-s + (−0.0846 + 0.0615i)14-s + (−0.799 − 0.580i)16-s + (0.229 − 0.0747i)17-s − 0.0211i·18-s + (−0.0957 − 0.294i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.744 + 0.667i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.744 + 0.667i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(375\)    =    \(3 \cdot 5^{3}\)
Sign: $-0.744 + 0.667i$
Analytic conductor: \(2.99439\)
Root analytic conductor: \(1.73043\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{375} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 375,\ (\ :1/2),\ -0.744 + 0.667i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.314602 - 0.822847i\)
\(L(\frac12)\) \(\approx\) \(0.314602 - 0.822847i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.0527 + 0.0726i)T + (-0.618 + 1.90i)T^{2} \)
7 \( 1 + 4.36iT - 7T^{2} \)
11 \( 1 + (3.55 - 2.58i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (1.16 - 1.60i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (-0.948 + 0.308i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + (0.417 + 1.28i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (1.38 + 1.90i)T + (-7.10 + 21.8i)T^{2} \)
29 \( 1 + (-2.46 + 7.58i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (1.13 + 3.49i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-0.844 + 1.16i)T + (-11.4 - 35.1i)T^{2} \)
41 \( 1 + (-4.83 - 3.51i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 2.68iT - 43T^{2} \)
47 \( 1 + (10.4 + 3.38i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (-10.5 - 3.41i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (-5.41 - 3.93i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-7.64 + 5.55i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-12.2 + 3.99i)T + (54.2 - 39.3i)T^{2} \)
71 \( 1 + (-2.26 + 6.96i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-0.249 - 0.343i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (-1.96 + 6.04i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (0.700 - 0.227i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 + (7.91 - 5.75i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (0.0320 + 0.0104i)T + (78.4 + 57.0i)T^{2} \)
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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

−10.93203561818611873597499605155, −10.12100680237313570583128655079, −9.741290231776252694309509659291, −7.910920954962770196203564639160, −7.13650097290571625901255078075, −6.32616439081726435721836889067, −5.07608172479909983332072720354, −4.24851614668633858452155187358, −2.22279529952755982888277620847, −0.61523527049292707819620435488, 2.43138284870306641310642454450, 3.42311126672887840486811160246, 5.14851195569082457732201135947, 5.79065739353895673497730329101, 7.00044252381233850063256414862, 8.205129498245900366388066061971, 8.701762684982698830001038430707, 9.955567369430438780919188584159, 11.03289424548067039038104976099, 11.75320427978716280907914285771

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