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Properties

Label 1-75-75.56-r1-0-0
Degree $1$
Conductor $75$
Sign $0.0627 - 0.998i$
Analytic cond. $8.05986$
Root an. cond. $8.05986$
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  + (0.809 − 0.587i)2-s + (0.309 − 0.951i)4-s + 7-s + (−0.309 − 0.951i)8-s + (0.809 − 0.587i)11-s + (−0.809 − 0.587i)13-s + (0.809 − 0.587i)14-s + (−0.809 − 0.587i)16-s + (−0.309 − 0.951i)17-s + (0.309 + 0.951i)19-s + (0.309 − 0.951i)22-s + (0.809 − 0.587i)23-s − 26-s + (0.309 − 0.951i)28-s + (−0.309 + 0.951i)29-s + ⋯
L(s)  = 1  + (0.809 − 0.587i)2-s + (0.309 − 0.951i)4-s + 7-s + (−0.309 − 0.951i)8-s + (0.809 − 0.587i)11-s + (−0.809 − 0.587i)13-s + (0.809 − 0.587i)14-s + (−0.809 − 0.587i)16-s + (−0.309 − 0.951i)17-s + (0.309 + 0.951i)19-s + (0.309 − 0.951i)22-s + (0.809 − 0.587i)23-s − 26-s + (0.309 − 0.951i)28-s + (−0.309 + 0.951i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0627 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0627 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $0.0627 - 0.998i$
Analytic conductor: \(8.05986\)
Root analytic conductor: \(8.05986\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (56, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 75,\ (1:\ ),\ 0.0627 - 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.996325396 - 1.874674329i\)
\(L(\frac12)\) \(\approx\) \(1.996325396 - 1.874674329i\)
\(L(1)\) \(\approx\) \(1.599498581 - 0.8509635120i\)
\(L(1)\) \(\approx\) \(1.599498581 - 0.8509635120i\)

Euler product

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

Imaginary part of the first few zeros on the critical line

−31.29409625511506803948156990957, −30.61680949277836487430652570077, −29.62189920563254641005971046024, −28.11057268576252833602800149405, −26.88312179186039641393422930765, −25.8537278793337975229734604197, −24.54458222050116063668279311626, −24.03656851830482884377707058528, −22.68694513485175305701640152675, −21.71479154722156042997723435612, −20.74254461874352251304228532071, −19.43583435718875475742972372783, −17.542358468276466534607929048298, −17.06239348929827036215311424873, −15.3624046098181432318689526360, −14.64781809035259079548577537167, −13.50811124040876612838263776882, −12.1322022744080272692302678570, −11.207544382510861583864686964773, −9.19622246362116727541383888932, −7.78007233239186372638372240647, −6.66963920491949206928636971002, −5.09561414300904787840868760393, −4.064036657239392203646570385655, −2.10377499584930417717223421599, 1.21036398261233875901702064928, 2.9005079014347485033513058392, 4.48361606531549343966401613361, 5.61368832940562165479106766333, 7.23736760192751313395663446946, 9.00623954568903237201045854370, 10.52051037661170176605383083573, 11.55988477516419032240285932934, 12.57542993808367665196050355637, 14.10677337944532839483216578679, 14.66895643545789445313713529650, 16.15803527640049457507775759110, 17.684059134375126980962391716934, 18.95937152388132443099241894952, 20.12840167660519465458366930890, 21.034937609875021059934452686579, 22.12877546176181087177962982313, 23.04539105673167620198155809541, 24.47436808857309265195795491656, 24.85197183130739087321676082681, 27.02650024561109229652436306506, 27.617920927504250744600142774683, 29.09559872130748778159302035445, 29.851579676549375571526185416531, 30.9048994074797685904766924663

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