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Properties

Label 2-975-13.9-c1-0-29
Degree $2$
Conductor $975$
Sign $0.794 + 0.607i$
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  + (−0.780 + 1.35i)2-s + (0.5 − 0.866i)3-s + (−0.219 − 0.379i)4-s + (0.780 + 1.35i)6-s + (0.280 + 0.486i)7-s − 2.43·8-s + (−0.499 − 0.866i)9-s + (1 − 1.73i)11-s − 0.438·12-s + (−0.5 − 3.57i)13-s − 0.876·14-s + (2.34 − 4.05i)16-s + (−0.780 − 1.35i)17-s + 1.56·18-s + (−3.56 − 6.16i)19-s + ⋯
L(s)  = 1  + (−0.552 + 0.956i)2-s + (0.288 − 0.499i)3-s + (−0.109 − 0.189i)4-s + (0.318 + 0.552i)6-s + (0.106 + 0.183i)7-s − 0.862·8-s + (−0.166 − 0.288i)9-s + (0.301 − 0.522i)11-s − 0.126·12-s + (−0.138 − 0.990i)13-s − 0.234·14-s + (0.585 − 1.01i)16-s + (−0.189 − 0.327i)17-s + 0.368·18-s + (−0.817 − 1.41i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.794 + 0.607i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.794 + 0.607i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $0.794 + 0.607i$
Analytic conductor: \(7.78541\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ 0.794 + 0.607i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.927774 - 0.314000i\)
\(L(\frac12)\) \(\approx\) \(0.927774 - 0.314000i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 \)
13 \( 1 + (0.5 + 3.57i)T \)
good2 \( 1 + (0.780 - 1.35i)T + (-1 - 1.73i)T^{2} \)
7 \( 1 + (-0.280 - 0.486i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1 + 1.73i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (0.780 + 1.35i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.56 + 6.16i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1 + 1.73i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.34 - 5.78i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 2.56T + 31T^{2} \)
37 \( 1 + (-3.78 + 6.54i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.780 + 1.35i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.28 - 3.95i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 8.24T + 47T^{2} \)
53 \( 1 - 0.684T + 53T^{2} \)
59 \( 1 + (-1.43 - 2.49i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.93 + 3.35i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.28 + 3.95i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (7 + 12.1i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 10.1T + 73T^{2} \)
79 \( 1 - 5.43T + 79T^{2} \)
83 \( 1 - 0.876T + 83T^{2} \)
89 \( 1 + (2.43 - 4.22i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (4.28 + 7.41i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.463793759007912096707734800168, −8.855032435440334475859135793562, −8.219506751840500466527780067400, −7.41771464103934915163795519906, −6.70783643648768649749139802353, −5.95013017843407542749611852900, −4.93296840062150340868412944574, −3.38277832236644816267337562490, −2.44628411343413607083807512047, −0.52148446564148092987501816469, 1.52368919756866851052166615222, 2.42124580708980020038662580469, 3.72383653396687762720485066821, 4.44311418178179329671002752566, 5.82538229698854230968467569620, 6.66957099178080529037733587259, 7.919704483926122371542370081006, 8.693718192465772125186650293138, 9.598527066270845040992663827794, 9.952576489296247143006947107776

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