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Properties

Label 2-975-195.173-c1-0-24
Degree $2$
Conductor $975$
Sign $-0.812 + 0.582i$
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  + (0.496 + 1.85i)2-s + (0.676 + 1.59i)3-s + (−1.45 + 0.837i)4-s + (−2.61 + 2.04i)6-s + (−0.527 + 1.97i)7-s + (0.440 + 0.440i)8-s + (−2.08 + 2.15i)9-s + (−0.0324 + 0.0561i)11-s + (−2.31 − 1.74i)12-s + (−3.39 − 1.22i)13-s − 3.91·14-s + (−2.27 + 3.93i)16-s + (1.31 − 4.88i)17-s + (−5.02 − 2.79i)18-s + (1.37 + 2.38i)19-s + ⋯
L(s)  = 1  + (0.350 + 1.30i)2-s + (0.390 + 0.920i)3-s + (−0.725 + 0.418i)4-s + (−1.06 + 0.834i)6-s + (−0.199 + 0.744i)7-s + (0.155 + 0.155i)8-s + (−0.694 + 0.719i)9-s + (−0.00977 + 0.0169i)11-s + (−0.668 − 0.504i)12-s + (−0.940 − 0.338i)13-s − 1.04·14-s + (−0.567 + 0.983i)16-s + (0.317 − 1.18i)17-s + (−1.18 − 0.657i)18-s + (0.315 + 0.546i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.812 + 0.582i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.812 + 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $-0.812 + 0.582i$
Analytic conductor: \(7.78541\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (368, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ -0.812 + 0.582i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.570122 - 1.77359i\)
\(L(\frac12)\) \(\approx\) \(0.570122 - 1.77359i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.676 - 1.59i)T \)
5 \( 1 \)
13 \( 1 + (3.39 + 1.22i)T \)
good2 \( 1 + (-0.496 - 1.85i)T + (-1.73 + i)T^{2} \)
7 \( 1 + (0.527 - 1.97i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (0.0324 - 0.0561i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1.31 + 4.88i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-1.37 - 2.38i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.36 - 5.10i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (1.40 - 2.42i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 0.312iT - 31T^{2} \)
37 \( 1 + (-1.21 + 0.326i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-5.66 + 9.81i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.53 - 9.44i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (-0.638 + 0.638i)T - 47iT^{2} \)
53 \( 1 + (-5.43 + 5.43i)T - 53iT^{2} \)
59 \( 1 + (5.86 - 3.38i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.38 + 2.39i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (10.5 - 2.83i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-6.90 - 11.9i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-7.38 + 7.38i)T - 73iT^{2} \)
79 \( 1 + 5.34iT - 79T^{2} \)
83 \( 1 + (2.60 + 2.60i)T + 83iT^{2} \)
89 \( 1 + (-12.5 - 7.22i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.90 + 7.12i)T + (-84.0 - 48.5i)T^{2} \)
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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

−10.25453403704359202396573322321, −9.414286557512363637995579760708, −8.854157490422197918028645181811, −7.74203240598988368163867098675, −7.33384720848404108852152704526, −5.99646925837565437942019573961, −5.33464566742802340419128264921, −4.75016549958343394238494383610, −3.48286715937177507339589974801, −2.38312665864537757697561488014, 0.71767021039594084679865421321, 1.93711538378849418389058439808, 2.84120046581741172066148932460, 3.80261446319216694192230629962, 4.73214482538679614729441536846, 6.16770043210411420674302334824, 7.09004892035108975742690818206, 7.72598175496206737699570510531, 8.839049704751188506309068016082, 9.730990382629484352988460056544

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