↑

Properties

Label 2-290-145.102-c1-0-12
Degree $2$
Conductor $290$
Sign $-0.670 + 0.742i$
Analytic cond. $2.31566$
Root an. cond. $1.52172$
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  + (−0.781 − 0.623i)2-s + (0.232 − 1.01i)3-s + (0.222 + 0.974i)4-s + (−1.62 − 1.53i)5-s + (−0.817 + 0.651i)6-s + (2.78 − 1.75i)7-s + (0.433 − 0.900i)8-s + (1.71 + 0.827i)9-s + (0.316 + 2.21i)10-s + (−3.10 − 1.08i)11-s + 1.04·12-s + (−1.61 − 0.563i)13-s + (−3.27 − 0.368i)14-s + (−1.94 + 1.30i)15-s + (−0.900 + 0.433i)16-s − 3.90i·17-s + ⋯
L(s)  = 1  + (−0.552 − 0.440i)2-s + (0.134 − 0.588i)3-s + (0.111 + 0.487i)4-s + (−0.727 − 0.685i)5-s + (−0.333 + 0.266i)6-s + (1.05 − 0.662i)7-s + (0.153 − 0.318i)8-s + (0.572 + 0.275i)9-s + (0.100 + 0.699i)10-s + (−0.934 − 0.327i)11-s + 0.301·12-s + (−0.446 − 0.156i)13-s + (−0.875 − 0.0985i)14-s + (−0.501 + 0.336i)15-s + (−0.225 + 0.108i)16-s − 0.948i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 290 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.670 + 0.742i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 290 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.670 + 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(290\)    =    \(2 \cdot 5 \cdot 29\)
Sign: $-0.670 + 0.742i$
Analytic conductor: \(2.31566\)
Root analytic conductor: \(1.52172\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{290} (247, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 290,\ (\ :1/2),\ -0.670 + 0.742i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.361833 - 0.814167i\)
\(L(\frac12)\) \(\approx\) \(0.361833 - 0.814167i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.781 + 0.623i)T \)
5 \( 1 + (1.62 + 1.53i)T \)
29 \( 1 + (5.30 + 0.952i)T \)
good3 \( 1 + (-0.232 + 1.01i)T + (-2.70 - 1.30i)T^{2} \)
7 \( 1 + (-2.78 + 1.75i)T + (3.03 - 6.30i)T^{2} \)
11 \( 1 + (3.10 + 1.08i)T + (8.60 + 6.85i)T^{2} \)
13 \( 1 + (1.61 + 0.563i)T + (10.1 + 8.10i)T^{2} \)
17 \( 1 + 3.90iT - 17T^{2} \)
19 \( 1 + (3.62 + 2.27i)T + (8.24 + 17.1i)T^{2} \)
23 \( 1 + (0.559 + 0.0630i)T + (22.4 + 5.11i)T^{2} \)
31 \( 1 + (-2.98 + 0.335i)T + (30.2 - 6.89i)T^{2} \)
37 \( 1 + (-6.47 - 3.11i)T + (23.0 + 28.9i)T^{2} \)
41 \( 1 + (-1.86 + 1.86i)T - 41iT^{2} \)
43 \( 1 + (0.281 + 0.352i)T + (-9.56 + 41.9i)T^{2} \)
47 \( 1 + (-5.66 + 2.72i)T + (29.3 - 36.7i)T^{2} \)
53 \( 1 + (-0.478 - 4.25i)T + (-51.6 + 11.7i)T^{2} \)
59 \( 1 + 0.857iT - 59T^{2} \)
61 \( 1 + (-1.49 + 0.938i)T + (26.4 - 54.9i)T^{2} \)
67 \( 1 + (-12.2 + 4.28i)T + (52.3 - 41.7i)T^{2} \)
71 \( 1 + (-6.68 - 13.8i)T + (-44.2 + 55.5i)T^{2} \)
73 \( 1 + (4.30 - 3.43i)T + (16.2 - 71.1i)T^{2} \)
79 \( 1 + (-8.83 + 3.09i)T + (61.7 - 49.2i)T^{2} \)
83 \( 1 + (-14.7 - 9.26i)T + (36.0 + 74.7i)T^{2} \)
89 \( 1 + (-0.484 - 4.29i)T + (-86.7 + 19.8i)T^{2} \)
97 \( 1 + (-1.11 - 4.88i)T + (-87.3 + 42.0i)T^{2} \)
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

−11.35043821349742338649963151488, −10.71508832510474607750300728903, −9.557364179621330568357637088508, −8.292576645856768761605946433822, −7.80922965588056789827181265146, −7.07223937515871725171569521250, −5.08436697320378382004583696376, −4.18644876086962659397194466303, −2.36225009991798605436042875717, −0.811288295315446688355112849935, 2.20019289190855371158220104125, 3.97848377914707694300101112514, 5.03405095582688620514122701637, 6.35723582754836824469276808189, 7.61425978671389968538450669722, 8.147562014177522380444586385442, 9.266004091870616456188088843445, 10.35699519466694667436767380506, 10.88251617511090809535869348145, 11.95841323483324132894400785784

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