↑

Properties

Label 2-952-952.781-c1-0-65
Degree $2$
Conductor $952$
Sign $0.726 + 0.687i$
Analytic cond. $7.60175$
Root an. cond. $2.75712$
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  + (−1.36 − 0.383i)2-s + (1.05 − 1.82i)3-s + (1.70 + 1.04i)4-s + (−0.211 − 0.366i)5-s + (−2.13 + 2.08i)6-s + (2.37 + 1.16i)7-s + (−1.92 − 2.07i)8-s + (−0.722 − 1.25i)9-s + (0.147 + 0.580i)10-s + (−1.23 + 2.13i)11-s + (3.70 − 2.01i)12-s + 1.83i·13-s + (−2.78 − 2.49i)14-s − 0.892·15-s + (1.81 + 3.56i)16-s + (2.27 + 3.43i)17-s + ⋯
L(s)  = 1  + (−0.962 − 0.271i)2-s + (0.608 − 1.05i)3-s + (0.852 + 0.522i)4-s + (−0.0947 − 0.164i)5-s + (−0.871 + 0.849i)6-s + (0.898 + 0.439i)7-s + (−0.678 − 0.734i)8-s + (−0.240 − 0.417i)9-s + (0.0466 + 0.183i)10-s + (−0.371 + 0.642i)11-s + (1.06 − 0.580i)12-s + 0.509i·13-s + (−0.745 − 0.666i)14-s − 0.230·15-s + (0.454 + 0.890i)16-s + (0.552 + 0.833i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 952 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.726 + 0.687i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 952 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.726 + 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(952\)    =    \(2^{3} \cdot 7 \cdot 17\)
Sign: $0.726 + 0.687i$
Analytic conductor: \(7.60175\)
Root analytic conductor: \(2.75712\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{952} (781, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 952,\ (\ :1/2),\ 0.726 + 0.687i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.34019 - 0.533430i\)
\(L(\frac12)\) \(\approx\) \(1.34019 - 0.533430i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.36 + 0.383i)T \)
7 \( 1 + (-2.37 - 1.16i)T \)
17 \( 1 + (-2.27 - 3.43i)T \)
good3 \( 1 + (-1.05 + 1.82i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.211 + 0.366i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.23 - 2.13i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 1.83iT - 13T^{2} \)
19 \( 1 + (-6.78 + 3.91i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.53 - 2.03i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 1.13T + 29T^{2} \)
31 \( 1 + (-4.95 - 2.86i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.84 + 6.66i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 5.93iT - 41T^{2} \)
43 \( 1 + 2.67iT - 43T^{2} \)
47 \( 1 + (-3.80 - 6.59i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-8.90 - 5.13i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (8.32 + 4.80i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.29 + 10.9i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.35 - 1.93i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 2.73iT - 71T^{2} \)
73 \( 1 + (0.639 + 0.369i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (12.6 - 7.29i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 7.35iT - 83T^{2} \)
89 \( 1 + (6.73 + 11.6i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 11.7iT - 97T^{2} \)
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

−9.793070397136763905409539075763, −8.904122865011542700200381602977, −8.227097889316225949012765358776, −7.59378417996731363188589046830, −7.04315961955364362065444526799, −5.86239012442850666867606863933, −4.56704363016428394306004502526, −3.01565437100000654117763287846, −2.06514379381227364431082985559, −1.23093266848456342577214234809, 1.08821527741085152282319399440, 2.77678098268547598372564789599, 3.64660751445082385757064304062, 5.00335764191446687625202975220, 5.71703347674240636054361938434, 7.14416745695744021041419095478, 7.84389173924333031342560164544, 8.490011669404853178098803308855, 9.315658952999841060972363566743, 10.23621910099834986831797478609

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