Properties

Label 2-75-15.14-c8-0-19
Degree $2$
Conductor $75$
Sign $0.557 - 0.830i$
Analytic cond. $30.5533$
Root an. cond. $5.52751$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 8.27·2-s + (39.9 + 70.4i)3-s − 187.·4-s + (−330. − 582. i)6-s − 860. i·7-s + 3.66e3·8-s + (−3.36e3 + 5.63e3i)9-s − 8.66e3i·11-s + (−7.50e3 − 1.32e4i)12-s − 1.84e4i·13-s + 7.11e3i·14-s + 1.76e4·16-s + 1.10e5·17-s + (2.78e4 − 4.65e4i)18-s + 7.12e4·19-s + ⋯
L(s)  = 1  − 0.516·2-s + (0.493 + 0.869i)3-s − 0.732·4-s + (−0.255 − 0.449i)6-s − 0.358i·7-s + 0.895·8-s + (−0.512 + 0.858i)9-s − 0.591i·11-s + (−0.361 − 0.637i)12-s − 0.646i·13-s + 0.185i·14-s + 0.269·16-s + 1.31·17-s + (0.264 − 0.443i)18-s + 0.546·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.557 - 0.830i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.557 - 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $0.557 - 0.830i$
Analytic conductor: \(30.5533\)
Root analytic conductor: \(5.52751\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (74, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :4),\ 0.557 - 0.830i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(1.23572 + 0.659087i\)
\(L(\frac12)\) \(\approx\) \(1.23572 + 0.659087i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-39.9 - 70.4i)T \)
5 \( 1 \)
good2 \( 1 + 8.27T + 256T^{2} \)
7 \( 1 + 860. iT - 5.76e6T^{2} \)
11 \( 1 + 8.66e3iT - 2.14e8T^{2} \)
13 \( 1 + 1.84e4iT - 8.15e8T^{2} \)
17 \( 1 - 1.10e5T + 6.97e9T^{2} \)
19 \( 1 - 7.12e4T + 1.69e10T^{2} \)
23 \( 1 + 3.69e5T + 7.83e10T^{2} \)
29 \( 1 + 3.06e5iT - 5.00e11T^{2} \)
31 \( 1 - 1.39e6T + 8.52e11T^{2} \)
37 \( 1 - 3.68e6iT - 3.51e12T^{2} \)
41 \( 1 - 5.05e6iT - 7.98e12T^{2} \)
43 \( 1 + 1.42e6iT - 1.16e13T^{2} \)
47 \( 1 - 7.79e5T + 2.38e13T^{2} \)
53 \( 1 - 7.55e6T + 6.22e13T^{2} \)
59 \( 1 - 2.08e7iT - 1.46e14T^{2} \)
61 \( 1 - 2.03e7T + 1.91e14T^{2} \)
67 \( 1 - 1.37e6iT - 4.06e14T^{2} \)
71 \( 1 - 4.19e6iT - 6.45e14T^{2} \)
73 \( 1 + 2.57e7iT - 8.06e14T^{2} \)
79 \( 1 + 1.07e7T + 1.51e15T^{2} \)
83 \( 1 + 1.63e7T + 2.25e15T^{2} \)
89 \( 1 - 5.98e7iT - 3.93e15T^{2} \)
97 \( 1 + 1.07e8iT - 7.83e15T^{2} \)
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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

−13.44846821305762106514337079824, −11.76476961912744948864922672245, −10.18983334153382108562529706846, −9.925655749257966413085966210562, −8.467587677597069720757844998385, −7.84046423785241237096581333252, −5.62353385878439937254131840001, −4.34746073731278581902106603999, −3.12994811010127223118151322198, −0.917052797171537186752059049765, 0.71612918867684630868696612329, 2.06248593846641406580194271879, 3.85416724926277883772712485259, 5.58357688479141289978764886987, 7.20163124770364817789333972166, 8.147712241705620641752570789476, 9.174448126437567983898879169530, 10.09942236525801468016855632298, 11.90212774433756893163486503310, 12.66079115619568870152602839170

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