↑

Properties

Label 2-975-195.164-c1-0-5
Degree $2$
Conductor $975$
Sign $0.981 - 0.188i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.978 − 0.978i)2-s + (−0.146 − 1.72i)3-s − 0.0848i·4-s + (−1.54 + 1.83i)6-s + (2.39 + 2.39i)7-s + (−2.04 + 2.04i)8-s + (−2.95 + 0.505i)9-s + (−3.70 − 3.70i)11-s + (−0.146 + 0.0124i)12-s + (−0.231 + 3.59i)13-s − 4.69i·14-s + 3.82·16-s + 6.45i·17-s + (3.38 + 2.39i)18-s + (2.23 + 2.23i)19-s + ⋯
L(s)  = 1  + (−0.691 − 0.691i)2-s + (−0.0845 − 0.996i)3-s − 0.0424i·4-s + (−0.630 + 0.747i)6-s + (0.906 + 0.906i)7-s + (−0.721 + 0.721i)8-s + (−0.985 + 0.168i)9-s + (−1.11 − 1.11i)11-s + (−0.0422 + 0.00358i)12-s + (−0.0641 + 0.997i)13-s − 1.25i·14-s + 0.955·16-s + 1.56i·17-s + (0.798 + 0.565i)18-s + (0.511 + 0.511i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.188i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.981 - 0.188i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $0.981 - 0.188i$
Analytic conductor: \(7.78541\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (749, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ 0.981 - 0.188i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.609175 + 0.0580770i\)
\(L(\frac12)\) \(\approx\) \(0.609175 + 0.0580770i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.146 + 1.72i)T \)
5 \( 1 \)
13 \( 1 + (0.231 - 3.59i)T \)
good2 \( 1 + (0.978 + 0.978i)T + 2iT^{2} \)
7 \( 1 + (-2.39 - 2.39i)T + 7iT^{2} \)
11 \( 1 + (3.70 + 3.70i)T + 11iT^{2} \)
17 \( 1 - 6.45iT - 17T^{2} \)
19 \( 1 + (-2.23 - 2.23i)T + 19iT^{2} \)
23 \( 1 + 3.42iT - 23T^{2} \)
29 \( 1 - 8.11iT - 29T^{2} \)
31 \( 1 + (2.54 + 2.54i)T + 31iT^{2} \)
37 \( 1 + (-2.23 - 2.23i)T + 37iT^{2} \)
41 \( 1 + (7.81 - 7.81i)T - 41iT^{2} \)
43 \( 1 + 7.98T + 43T^{2} \)
47 \( 1 + (2.03 - 2.03i)T - 47iT^{2} \)
53 \( 1 - 12.4T + 53T^{2} \)
59 \( 1 + (2.61 + 2.61i)T + 59iT^{2} \)
61 \( 1 - 1.52T + 61T^{2} \)
67 \( 1 + (5.93 - 5.93i)T - 67iT^{2} \)
71 \( 1 + (-4.09 + 4.09i)T - 71iT^{2} \)
73 \( 1 + (-0.226 - 0.226i)T + 73iT^{2} \)
79 \( 1 - 13.6T + 79T^{2} \)
83 \( 1 + (5.18 + 5.18i)T + 83iT^{2} \)
89 \( 1 + (2.62 + 2.62i)T + 89iT^{2} \)
97 \( 1 + (4.10 - 4.10i)T - 97iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.21950552924793094592363684605, −8.950840923141273601189517775364, −8.409081586803584578048108916805, −7.991529718314408908575411728985, −6.56637717195091628242453080335, −5.74290062984414016123302407449, −5.09616346393245356442794660947, −3.18646761319539531394005971003, −2.12841844583364591131086415460, −1.39206605610589419183910599642, 0.36736388671636630957121287703, 2.66300139959986487610875921480, 3.79086930625620828845771118697, 4.91855269211752841358529447809, 5.41970789327430068642265211738, 6.98653971418913843589831982308, 7.59407360624290759178604747614, 8.155869199635656417861605391880, 9.200693087338149131937493539441, 9.943908054416593911266642607036

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