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Properties

Label 2-975-195.89-c1-0-39
Degree $2$
Conductor $975$
Sign $-0.104 + 0.994i$
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  + (0.389 − 1.45i)2-s + (−1.71 − 0.239i)3-s + (−0.232 − 0.133i)4-s + (−1.01 + 2.40i)6-s + (−1.36 + 0.366i)7-s + (1.84 − 1.84i)8-s + (2.88 + 0.820i)9-s + (−1.06 + 3.97i)11-s + (0.366 + 0.285i)12-s + (0.232 − 3.59i)13-s + 2.12i·14-s + (−2.23 − 3.86i)16-s + (4.36 + 2.51i)17-s + (2.31 − 3.87i)18-s + (3.73 − i)19-s + ⋯
L(s)  = 1  + (0.275 − 1.02i)2-s + (−0.990 − 0.138i)3-s + (−0.116 − 0.0669i)4-s + (−0.415 + 0.980i)6-s + (−0.516 + 0.138i)7-s + (0.652 − 0.652i)8-s + (0.961 + 0.273i)9-s + (−0.321 + 1.19i)11-s + (0.105 + 0.0823i)12-s + (0.0643 − 0.997i)13-s + 0.569i·14-s + (−0.558 − 0.966i)16-s + (1.05 + 0.611i)17-s + (0.546 − 0.913i)18-s + (0.856 − 0.229i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.104 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.104 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $-0.104 + 0.994i$
Analytic conductor: \(7.78541\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (674, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ -0.104 + 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.973650 - 1.08098i\)
\(L(\frac12)\) \(\approx\) \(0.973650 - 1.08098i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.71 + 0.239i)T \)
5 \( 1 \)
13 \( 1 + (-0.232 + 3.59i)T \)
good2 \( 1 + (-0.389 + 1.45i)T + (-1.73 - i)T^{2} \)
7 \( 1 + (1.36 - 0.366i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (1.06 - 3.97i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (-4.36 - 2.51i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.73 + i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-6.20 + 3.58i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.46 + 2.46i)T + 31iT^{2} \)
37 \( 1 + (-1.40 + 5.23i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-5.42 - 1.45i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-1.09 + 1.90i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.25 + 4.25i)T - 47iT^{2} \)
53 \( 1 + 0.779T + 53T^{2} \)
59 \( 1 + (-2.90 + 0.779i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-3.5 + 6.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.73 + 1.53i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (0.779 + 2.90i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-0.901 - 0.901i)T + 73iT^{2} \)
79 \( 1 + 2T + 79T^{2} \)
83 \( 1 + (2.90 + 2.90i)T + 83iT^{2} \)
89 \( 1 + (-2.41 + 9.01i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-0.437 - 1.63i)T + (-84.0 + 48.5i)T^{2} \)
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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

−10.13858233379130098020199789228, −9.502060531657308788571871993426, −7.80595483938993324418142709569, −7.31109972403168873819314893105, −6.22561218661486468405842310880, −5.33173352052875052034126464544, −4.37586757356498095181006399252, −3.32940578925124549325236916421, −2.18976654477988402835890210912, −0.865164695145537842524509933040, 1.16192970564519131118015917327, 3.11257632645460103335218497723, 4.39600779072434378826417643791, 5.36117927060214303796125248131, 5.90537608978892167782553548171, 6.73349774078354972919216229841, 7.34569641676802476855356524769, 8.335803392251548875945262429517, 9.445977954732508458931033006199, 10.28603861060317129442682379594

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