↑

Properties

Label 2-74-37.10-c1-0-2
Degree $2$
Conductor $74$
Sign $0.957 + 0.289i$
Analytic cond. $0.590892$
Root an. cond. $0.768695$
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  + (0.5 − 0.866i)2-s + (1 + 1.73i)3-s + (−0.499 − 0.866i)4-s + (−1.5 − 2.59i)5-s + 1.99·6-s + (2 + 3.46i)7-s − 0.999·8-s + (−0.499 + 0.866i)9-s − 3·10-s − 6·11-s + (0.999 − 1.73i)12-s + (−1 − 1.73i)13-s + 3.99·14-s + (3 − 5.19i)15-s + (−0.5 + 0.866i)16-s + (−1.5 + 2.59i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.577 + 0.999i)3-s + (−0.249 − 0.433i)4-s + (−0.670 − 1.16i)5-s + 0.816·6-s + (0.755 + 1.30i)7-s − 0.353·8-s + (−0.166 + 0.288i)9-s − 0.948·10-s − 1.80·11-s + (0.288 − 0.499i)12-s + (−0.277 − 0.480i)13-s + 1.06·14-s + (0.774 − 1.34i)15-s + (−0.125 + 0.216i)16-s + (−0.363 + 0.630i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 + 0.289i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.957 + 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $0.957 + 0.289i$
Analytic conductor: \(0.590892\)
Root analytic conductor: \(0.768695\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{74} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 74,\ (\ :1/2),\ 0.957 + 0.289i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.12755 - 0.167072i\)
\(L(\frac12)\) \(\approx\) \(1.12755 - 0.167072i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (-5.5 - 2.59i)T \)
good3 \( 1 + (-1 - 1.73i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (1.5 + 2.59i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-2 - 3.46i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + 6T + 11T^{2} \)
13 \( 1 + (1 + 1.73i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.5 - 2.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1 + 1.73i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 6T + 23T^{2} \)
29 \( 1 - 3T + 29T^{2} \)
31 \( 1 - 2T + 31T^{2} \)
41 \( 1 + (1.5 + 2.59i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + 6T + 47T^{2} \)
53 \( 1 + (-3 + 5.19i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1 + 1.73i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-6 - 10.3i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 10T + 73T^{2} \)
79 \( 1 + (7 + 12.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3 - 5.19i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (1.5 - 2.59i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 13T + 97T^{2} \)
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

−14.95653443958464231055700798311, −13.15604242257151514758203168890, −12.45115738634829876929923446628, −11.26414665020741052750068267436, −10.08308586760505866467050837192, −8.766420929132927914421575909952, −8.243035394302099928905449408459, −5.25336430560591741725632318855, −4.62481559599403455827310685414, −2.79092090144508449658333842243, 2.84776382367424505759218468873, 4.68609982432455321402057069443, 6.89372335788240693191062248397, 7.47356788247230065397830730477, 8.165443290775452655722194638782, 10.40682458920880950055875691345, 11.33939819520528070453147229241, 12.92716226973792120455014388449, 13.70109124751105375943807488161, 14.46612310158116595310787215373

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