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Properties

Label 2-375-25.9-c1-0-0
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: $\chi_{375} (49, \cdot )$
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

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

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