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Properties

Label 2-975-195.119-c1-0-15
Degree $2$
Conductor $975$
Sign $0.655 + 0.754i$
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  + (−2.31 − 0.619i)2-s + (−1.16 − 1.28i)3-s + (3.23 + 1.86i)4-s + (1.89 + 3.68i)6-s + (0.366 + 1.36i)7-s + (−2.93 − 2.93i)8-s + (−0.292 + 2.98i)9-s + (−1.69 − 0.453i)11-s + (−1.36 − 6.31i)12-s + (−3.23 − 1.59i)13-s − 3.38i·14-s + (1.23 + 2.13i)16-s + (−1.85 − 1.07i)17-s + (2.52 − 6.72i)18-s + (0.267 + i)19-s + ⋯
L(s)  = 1  + (−1.63 − 0.438i)2-s + (−0.671 − 0.740i)3-s + (1.61 + 0.933i)4-s + (0.773 + 1.50i)6-s + (0.138 + 0.516i)7-s + (−1.03 − 1.03i)8-s + (−0.0975 + 0.995i)9-s + (−0.510 − 0.136i)11-s + (−0.394 − 1.82i)12-s + (−0.896 − 0.443i)13-s − 0.904i·14-s + (0.308 + 0.533i)16-s + (−0.450 − 0.260i)17-s + (0.595 − 1.58i)18-s + (0.0614 + 0.229i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.655 + 0.754i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.655 + 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $0.655 + 0.754i$
Analytic conductor: \(7.78541\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (899, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ 0.655 + 0.754i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.379812 - 0.173180i\)
\(L(\frac12)\) \(\approx\) \(0.379812 - 0.173180i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.16 + 1.28i)T \)
5 \( 1 \)
13 \( 1 + (3.23 + 1.59i)T \)
good2 \( 1 + (2.31 + 0.619i)T + (1.73 + i)T^{2} \)
7 \( 1 + (-0.366 - 1.36i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (1.69 + 0.453i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (1.85 + 1.07i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.267 - i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.79 - 2.76i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4.46 + 4.46i)T - 31iT^{2} \)
37 \( 1 + (-6.59 - 1.76i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (0.166 - 0.619i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (4.09 - 7.09i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-6.77 - 6.77i)T + 47iT^{2} \)
53 \( 1 - 4.62T + 53T^{2} \)
59 \( 1 + (1.23 + 4.62i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-3.5 + 6.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.26 - 8.46i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (-4.62 + 1.23i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (-6.09 + 6.09i)T - 73iT^{2} \)
79 \( 1 + 2T + 79T^{2} \)
83 \( 1 + (-1.23 + 1.23i)T - 83iT^{2} \)
89 \( 1 + (-9.70 - 2.60i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-12.5 + 3.36i)T + (84.0 - 48.5i)T^{2} \)
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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

−9.899262580951794678721384427888, −9.133206926554681042808956991997, −8.093250833187473527773375915975, −7.71261317169191911085007769643, −6.82772797087368890257139919810, −5.82300218619830583715148705917, −4.81419670534458682426971504099, −2.78967062854769308191624064276, −2.03354395362887425897871855948, −0.65169166326043814499484609849, 0.64302295125809918752020266373, 2.27522373252000279487955936796, 4.00233888563655551243401594303, 5.03350264497968702772562995143, 6.09159988355088224132726320104, 6.99883087637739660357859943634, 7.58523083501063785132275595689, 8.687548647428145797516295590621, 9.296196448108887159674925514239, 10.18896859573891714745185760308

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