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Properties

Label 2-975-195.164-c1-0-72
Degree $2$
Conductor $975$
Sign $0.240 + 0.970i$
Analytic cond. $7.78541$
Root an. cond. $2.79023$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.455 + 0.455i)2-s + (1.67 + 0.446i)3-s − 1.58i·4-s + (0.559 + 0.966i)6-s + (−2.89 − 2.89i)7-s + (1.63 − 1.63i)8-s + (2.60 + 1.49i)9-s + (−2.45 − 2.45i)11-s + (0.707 − 2.65i)12-s + (1.09 − 3.43i)13-s − 2.64i·14-s − 1.68·16-s + 6.48i·17-s + (0.504 + 1.86i)18-s + (−3.48 − 3.48i)19-s + ⋯
L(s)  = 1  + (0.322 + 0.322i)2-s + (0.966 + 0.257i)3-s − 0.792i·4-s + (0.228 + 0.394i)6-s + (−1.09 − 1.09i)7-s + (0.577 − 0.577i)8-s + (0.867 + 0.498i)9-s + (−0.741 − 0.741i)11-s + (0.204 − 0.765i)12-s + (0.304 − 0.952i)13-s − 0.705i·14-s − 0.420·16-s + 1.57i·17-s + (0.118 + 0.439i)18-s + (−0.798 − 0.798i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.240 + 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.240 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $0.240 + 0.970i$
Analytic conductor: \(7.78541\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (749, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ 0.240 + 0.970i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.65131 - 1.29204i\)
\(L(\frac12)\) \(\approx\) \(1.65131 - 1.29204i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.67 - 0.446i)T \)
5 \( 1 \)
13 \( 1 + (-1.09 + 3.43i)T \)
good2 \( 1 + (-0.455 - 0.455i)T + 2iT^{2} \)
7 \( 1 + (2.89 + 2.89i)T + 7iT^{2} \)
11 \( 1 + (2.45 + 2.45i)T + 11iT^{2} \)
17 \( 1 - 6.48iT - 17T^{2} \)
19 \( 1 + (3.48 + 3.48i)T + 19iT^{2} \)
23 \( 1 + 2.39iT - 23T^{2} \)
29 \( 1 + 0.0728iT - 29T^{2} \)
31 \( 1 + (3.16 + 3.16i)T + 31iT^{2} \)
37 \( 1 + (-6.34 - 6.34i)T + 37iT^{2} \)
41 \( 1 + (-2.12 + 2.12i)T - 41iT^{2} \)
43 \( 1 - 3.53T + 43T^{2} \)
47 \( 1 + (-8.05 + 8.05i)T - 47iT^{2} \)
53 \( 1 - 9.32T + 53T^{2} \)
59 \( 1 + (-3.31 - 3.31i)T + 59iT^{2} \)
61 \( 1 + 1.29T + 61T^{2} \)
67 \( 1 + (0.922 - 0.922i)T - 67iT^{2} \)
71 \( 1 + (-2.96 + 2.96i)T - 71iT^{2} \)
73 \( 1 + (-5.91 - 5.91i)T + 73iT^{2} \)
79 \( 1 + 9.54T + 79T^{2} \)
83 \( 1 + (2.62 + 2.62i)T + 83iT^{2} \)
89 \( 1 + (-7.03 - 7.03i)T + 89iT^{2} \)
97 \( 1 + (6.77 - 6.77i)T - 97iT^{2} \)
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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

−10.05203250421309898689132798382, −8.982897954733154754786855298291, −8.168025470461473298000871587657, −7.25815234514854146208833263901, −6.39797999864311182743978912439, −5.59328575417238983763016413905, −4.29815301912412187451315860178, −3.65868549735631062108367540772, −2.47910088169081721561446788887, −0.75748990888561781772605653449, 2.16685368116662949986665688597, 2.69928254406333777528642145434, 3.68565743655995977032291254856, 4.62310506150571223878330040943, 5.93331435635664113523886198958, 7.07685599325736165642088543030, 7.59343765292586520024187434550, 8.699422571554047419899914214390, 9.213994751744466040470753149408, 9.907984283672081751635805710517

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