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Properties

Label 2-75-3.2-c2-0-0
Degree $2$
Conductor $75$
Sign $-0.833 + 0.552i$
Analytic cond. $2.04360$
Root an. cond. $1.42954$
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  + 3.31i·2-s + (−2.5 + 1.65i)3-s − 7·4-s + (−5.5 − 8.29i)6-s − 9.94i·8-s + (3.5 − 8.29i)9-s + 16.5i·11-s + (17.5 − 11.6i)12-s − 10·13-s + 5.00·16-s + 3.31i·17-s + (27.4 + 11.6i)18-s + 7·19-s − 55.0·22-s + 19.8i·23-s + (16.5 + 24.8i)24-s + ⋯
L(s)  = 1  + 1.65i·2-s + (−0.833 + 0.552i)3-s − 1.75·4-s + (−0.916 − 1.38i)6-s − 1.24i·8-s + (0.388 − 0.921i)9-s + 1.50i·11-s + (1.45 − 0.967i)12-s − 0.769·13-s + 0.312·16-s + 0.195i·17-s + (1.52 + 0.644i)18-s + 0.368·19-s − 2.50·22-s + 0.865i·23-s + (0.687 + 1.03i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.833 + 0.552i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.833 + 0.552i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $-0.833 + 0.552i$
Analytic conductor: \(2.04360\)
Root analytic conductor: \(1.42954\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :1),\ -0.833 + 0.552i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.212652 - 0.705287i\)
\(L(\frac12)\) \(\approx\) \(0.212652 - 0.705287i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.5 - 1.65i)T \)
5 \( 1 \)
good2 \( 1 - 3.31iT - 4T^{2} \)
7 \( 1 + 49T^{2} \)
11 \( 1 - 16.5iT - 121T^{2} \)
13 \( 1 + 10T + 169T^{2} \)
17 \( 1 - 3.31iT - 289T^{2} \)
19 \( 1 - 7T + 361T^{2} \)
23 \( 1 - 19.8iT - 529T^{2} \)
29 \( 1 - 33.1iT - 841T^{2} \)
31 \( 1 - 42T + 961T^{2} \)
37 \( 1 - 40T + 1.36e3T^{2} \)
41 \( 1 + 16.5iT - 1.68e3T^{2} \)
43 \( 1 - 50T + 1.84e3T^{2} \)
47 \( 1 + 46.4iT - 2.20e3T^{2} \)
53 \( 1 + 46.4iT - 2.80e3T^{2} \)
59 \( 1 - 66.3iT - 3.48e3T^{2} \)
61 \( 1 + 8T + 3.72e3T^{2} \)
67 \( 1 + 45T + 4.48e3T^{2} \)
71 \( 1 - 33.1iT - 5.04e3T^{2} \)
73 \( 1 - 35T + 5.32e3T^{2} \)
79 \( 1 - 12T + 6.24e3T^{2} \)
83 \( 1 - 69.6iT - 6.88e3T^{2} \)
89 \( 1 + 149. iT - 7.92e3T^{2} \)
97 \( 1 - 70T + 9.40e3T^{2} \)
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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

−15.17504411102317779283812417356, −14.41943868067291434002100009519, −12.91579344036196336152965421658, −11.78371664995541256777090216274, −10.13071473858409237880329163208, −9.235359735038001133082341022135, −7.59111921228887151435526646833, −6.69220502956559773239392690377, −5.37585889208735346786348859705, −4.44837874626504781478251437398, 0.72507732653626953368632610725, 2.70426003703818706466712273760, 4.59375851392575445562101445835, 6.17892388696250017207784049152, 8.031786887265594906957224790805, 9.573970437635075759008966940938, 10.71170401709734369883665800005, 11.47019098443622841110943855599, 12.26707260963880281144351454594, 13.25897015978267317957162402064

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