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Properties

Label 2-375-75.11-c2-0-24
Degree $2$
Conductor $375$
Sign $-0.417 - 0.908i$
Analytic cond. $10.2180$
Root an. cond. $3.19656$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.37 + 1.88i)2-s + (−1.35 + 2.67i)3-s + (−0.446 + 1.37i)4-s + (−6.91 + 1.11i)6-s + 12.5·7-s + (5.66 − 1.84i)8-s + (−5.32 − 7.25i)9-s + (7.59 + 10.4i)11-s + (−3.07 − 3.06i)12-s + (4.02 + 2.92i)13-s + (17.2 + 23.7i)14-s + (15.9 + 11.5i)16-s + (−10.2 + 3.32i)17-s + (6.39 − 20.0i)18-s + (1.29 + 3.99i)19-s + ⋯
L(s)  = 1  + (0.685 + 0.944i)2-s + (−0.451 + 0.892i)3-s + (−0.111 + 0.343i)4-s + (−1.15 + 0.185i)6-s + 1.79·7-s + (0.708 − 0.230i)8-s + (−0.591 − 0.806i)9-s + (0.690 + 0.950i)11-s + (−0.256 − 0.255i)12-s + (0.309 + 0.224i)13-s + (1.23 + 1.69i)14-s + (0.995 + 0.723i)16-s + (−0.602 + 0.195i)17-s + (0.355 − 1.11i)18-s + (0.0682 + 0.210i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.417 - 0.908i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.417 - 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(375\)    =    \(3 \cdot 5^{3}\)
Sign: $-0.417 - 0.908i$
Analytic conductor: \(10.2180\)
Root analytic conductor: \(3.19656\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{375} (176, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 375,\ (\ :1),\ -0.417 - 0.908i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.44491 + 2.25501i\)
\(L(\frac12)\) \(\approx\) \(1.44491 + 2.25501i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.35 - 2.67i)T \)
5 \( 1 \)
good2 \( 1 + (-1.37 - 1.88i)T + (-1.23 + 3.80i)T^{2} \)
7 \( 1 - 12.5T + 49T^{2} \)
11 \( 1 + (-7.59 - 10.4i)T + (-37.3 + 115. i)T^{2} \)
13 \( 1 + (-4.02 - 2.92i)T + (52.2 + 160. i)T^{2} \)
17 \( 1 + (10.2 - 3.32i)T + (233. - 169. i)T^{2} \)
19 \( 1 + (-1.29 - 3.99i)T + (-292. + 212. i)T^{2} \)
23 \( 1 + (12.8 + 17.6i)T + (-163. + 503. i)T^{2} \)
29 \( 1 + (26.0 + 8.47i)T + (680. + 494. i)T^{2} \)
31 \( 1 + (8.64 + 26.5i)T + (-777. + 564. i)T^{2} \)
37 \( 1 + (0.0279 + 0.0202i)T + (423. + 1.30e3i)T^{2} \)
41 \( 1 + (44.3 - 61.1i)T + (-519. - 1.59e3i)T^{2} \)
43 \( 1 - 43.3T + 1.84e3T^{2} \)
47 \( 1 + (4.67 + 1.52i)T + (1.78e3 + 1.29e3i)T^{2} \)
53 \( 1 + (1.62 + 0.528i)T + (2.27e3 + 1.65e3i)T^{2} \)
59 \( 1 + (28.0 - 38.6i)T + (-1.07e3 - 3.31e3i)T^{2} \)
61 \( 1 + (5.28 - 3.83i)T + (1.14e3 - 3.53e3i)T^{2} \)
67 \( 1 + (1.27 + 3.93i)T + (-3.63e3 + 2.63e3i)T^{2} \)
71 \( 1 + (51.8 + 16.8i)T + (4.07e3 + 2.96e3i)T^{2} \)
73 \( 1 + (-28.9 + 21.0i)T + (1.64e3 - 5.06e3i)T^{2} \)
79 \( 1 + (5.27 - 16.2i)T + (-5.04e3 - 3.66e3i)T^{2} \)
83 \( 1 + (-54.4 + 17.6i)T + (5.57e3 - 4.04e3i)T^{2} \)
89 \( 1 + (44.4 + 61.1i)T + (-2.44e3 + 7.53e3i)T^{2} \)
97 \( 1 + (-9.23 + 28.4i)T + (-7.61e3 - 5.53e3i)T^{2} \)
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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

−11.36944855293423110806037075701, −10.70255073814523464377462663468, −9.657838718439994840821967685873, −8.512786102850701663695169196166, −7.55327119628227398541174167392, −6.43532411724498847921289135246, −5.52351998940058350865682729985, −4.50725006280913849298312917999, −4.19335292465064193417337519598, −1.69981009666120801455152825393, 1.23648003806128579428376082025, 2.12503331668852685370032501545, 3.66488781435742019085678342575, 4.90728013596335652947230097713, 5.69869553145979050271266915322, 7.15391944807769616894780967167, 8.011020566774301049662935987890, 8.831031335461922595004773402486, 10.72063744486161855461839764730, 11.13476842777693334565798761647

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