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Properties

Label 2-975-195.149-c1-0-32
Degree $2$
Conductor $975$
Sign $-0.838 - 0.545i$
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 + (0.650 + 1.60i)3-s + (−0.232 + 0.133i)4-s + (−2.08 + 1.57i)6-s + (1.36 + 0.366i)7-s + (1.84 + 1.84i)8-s + (−2.15 + 2.08i)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 + (−3.87 − 2.31i)18-s + (3.73 + i)19-s + ⋯
L(s)  = 1  + (0.275 + 1.02i)2-s + (0.375 + 0.926i)3-s + (−0.116 + 0.0669i)4-s + (−0.849 + 0.641i)6-s + (0.516 + 0.138i)7-s + (0.652 + 0.652i)8-s + (−0.717 + 0.696i)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 + (−0.913 − 0.546i)18-s + (0.856 + 0.229i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.838 - 0.545i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.838 - 0.545i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $-0.838 - 0.545i$
Analytic conductor: \(7.78541\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ -0.838 - 0.545i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.706941 + 2.38464i\)
\(L(\frac12)\) \(\approx\) \(0.706941 + 2.38464i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.650 - 1.60i)T \)
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

−10.12561918196256294119575297263, −9.575708706528318367381041646035, −8.494881324052720634246160159377, −7.66027164497169688982869755013, −7.22257188082289381156808411681, −5.73361803073747475771842242791, −5.28448461908474925346581719701, −4.45375659307153837495144200896, −3.27496464565921194916625474788, −1.91753287068196516598806446686, 1.11630531034486236733417747307, 1.93166723806048051554732776823, 3.21254773800823472425922386035, 3.78260630840015414243835592584, 5.24106105874992196555762441533, 6.33905749689947852185124748445, 7.22797465458610591243053453138, 7.960257743893107758660476702629, 8.871767268574085692834087269181, 9.720661226133665705837607306713

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