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Properties

Label 2-1875-75.29-c0-0-7
Degree $2$
Conductor $1875$
Sign $0.770 + 0.637i$
Analytic cond. $0.935746$
Root an. cond. $0.967340$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.587 − 0.809i)3-s + (0.809 + 0.587i)4-s − 0.618i·7-s + (−0.309 − 0.951i)9-s + (0.951 − 0.309i)12-s + (1.53 − 0.5i)13-s + (0.309 + 0.951i)16-s + (−1.30 + 0.951i)19-s + (−0.500 − 0.363i)21-s + (−0.951 − 0.309i)27-s + (0.363 − 0.500i)28-s + (−0.5 + 0.363i)31-s + (0.309 − 0.951i)36-s + (−1.53 + 0.5i)37-s + (0.5 − 1.53i)39-s + ⋯
L(s)  = 1  + (0.587 − 0.809i)3-s + (0.809 + 0.587i)4-s − 0.618i·7-s + (−0.309 − 0.951i)9-s + (0.951 − 0.309i)12-s + (1.53 − 0.5i)13-s + (0.309 + 0.951i)16-s + (−1.30 + 0.951i)19-s + (−0.500 − 0.363i)21-s + (−0.951 − 0.309i)27-s + (0.363 − 0.500i)28-s + (−0.5 + 0.363i)31-s + (0.309 − 0.951i)36-s + (−1.53 + 0.5i)37-s + (0.5 − 1.53i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.770 + 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.770 + 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1875\)    =    \(3 \cdot 5^{4}\)
Sign: $0.770 + 0.637i$
Analytic conductor: \(0.935746\)
Root analytic conductor: \(0.967340\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1875} (749, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1875,\ (\ :0),\ 0.770 + 0.637i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.695464957\)
\(L(\frac12)\) \(\approx\) \(1.695464957\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.587 + 0.809i)T \)
5 \( 1 \)
good2 \( 1 + (-0.809 - 0.587i)T^{2} \)
7 \( 1 + 0.618iT - T^{2} \)
11 \( 1 + (0.809 + 0.587i)T^{2} \)
13 \( 1 + (-1.53 + 0.5i)T + (0.809 - 0.587i)T^{2} \)
17 \( 1 + (0.309 - 0.951i)T^{2} \)
19 \( 1 + (1.30 - 0.951i)T + (0.309 - 0.951i)T^{2} \)
23 \( 1 + (-0.809 - 0.587i)T^{2} \)
29 \( 1 + (-0.309 - 0.951i)T^{2} \)
31 \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \)
37 \( 1 + (1.53 - 0.5i)T + (0.809 - 0.587i)T^{2} \)
41 \( 1 + (0.809 - 0.587i)T^{2} \)
43 \( 1 + 1.61iT - T^{2} \)
47 \( 1 + (0.309 + 0.951i)T^{2} \)
53 \( 1 + (0.309 + 0.951i)T^{2} \)
59 \( 1 + (0.809 - 0.587i)T^{2} \)
61 \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \)
67 \( 1 + (-0.363 - 0.5i)T + (-0.309 + 0.951i)T^{2} \)
71 \( 1 + (-0.309 - 0.951i)T^{2} \)
73 \( 1 + (-0.587 - 0.190i)T + (0.809 + 0.587i)T^{2} \)
79 \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \)
83 \( 1 + (0.309 - 0.951i)T^{2} \)
89 \( 1 + (0.809 + 0.587i)T^{2} \)
97 \( 1 + (0.363 - 0.5i)T + (-0.309 - 0.951i)T^{2} \)
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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

−8.952664253373064024502681254337, −8.377433676952144338937357727943, −7.85222485413483423116955888184, −6.93131430814931482509878539484, −6.47076885798098651715884181279, −5.59656060646979222424092467627, −3.80988417751737619499013416945, −3.57581295295140894388646683057, −2.29833845798923823362260250907, −1.38014072890866274486869579556, 1.73872841089749654340073733757, 2.59395716920068404103478552417, 3.60576401646659514413972763585, 4.56830205196087352184927622167, 5.54352531577112260075978216585, 6.24603497359428433789404141246, 7.05382115478427860382521822736, 8.179593299426876601082867547904, 8.837566921150861938477894180746, 9.390384338638181464168014582578

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