Properties

Degree 2
Conductor 61
Sign $-0.855 + 0.518i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.5 + 0.866i)2-s − 2·3-s + (0.5 − 0.866i)4-s + (−1.5 − 2.59i)5-s + (3 − 1.73i)6-s + (−3 + 1.73i)7-s − 1.73i·8-s + 9-s + (4.5 + 2.59i)10-s + 3.46i·11-s + (−1 + 1.73i)12-s + (1 + 1.73i)13-s + (3 − 5.19i)14-s + (3 + 5.19i)15-s + (2.49 + 4.33i)16-s + (−6 − 3.46i)17-s + ⋯
L(s)  = 1  + (−1.06 + 0.612i)2-s − 1.15·3-s + (0.250 − 0.433i)4-s + (−0.670 − 1.16i)5-s + (1.22 − 0.707i)6-s + (−1.13 + 0.654i)7-s − 0.612i·8-s + 0.333·9-s + (1.42 + 0.821i)10-s + 1.04i·11-s + (−0.288 + 0.500i)12-s + (0.277 + 0.480i)13-s + (0.801 − 1.38i)14-s + (0.774 + 1.34i)15-s + (0.624 + 1.08i)16-s + (−1.45 − 0.840i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.855 + 0.518i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.855 + 0.518i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(61\)
\( \varepsilon \)  =  $-0.855 + 0.518i$
motivic weight  =  \(1\)
character  :  $\chi_{61} (14, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  1
Selberg data  =  $(2,\ 61,\ (\ :1/2),\ -0.855 + 0.518i)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \neq 61$,\(F_p(T)\) is a polynomial of degree 2. If $p = 61$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad61 \( 1 + (0.5 + 7.79i)T \)
good2 \( 1 + (1.5 - 0.866i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + 2T + 3T^{2} \)
5 \( 1 + (1.5 + 2.59i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (3 - 1.73i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 - 3.46iT - 11T^{2} \)
13 \( 1 + (-1 - 1.73i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (6 + 3.46i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1 + 1.73i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + (1.5 + 0.866i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (9 + 5.19i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 1.73iT - 37T^{2} \)
41 \( 1 - 3T + 41T^{2} \)
43 \( 1 + (-3 + 1.73i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (6 - 10.3i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 5.19iT - 53T^{2} \)
59 \( 1 + (3 - 1.73i)T + (29.5 - 51.0i)T^{2} \)
67 \( 1 + (3 - 1.73i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (9 + 5.19i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-3.5 + 6.06i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (9 - 5.19i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (3 + 5.19i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 15.5iT - 89T^{2} \)
97 \( 1 + (0.5 - 0.866i)T + (-48.5 - 84.0i)T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−15.45398636081071759679123171310, −13.00669108642035085648456205411, −12.35773274924212323012558365673, −11.22175428149875369070283455099, −9.486771794996276417784994054962, −8.934124914792456645089787817877, −7.32351152279063211893662906148, −6.17446311783826414281828765443, −4.55343076693636250565205258201, 0, 3.39144853322135670810655197720, 5.92144739334591805205480698882, 7.08316854214303469120456888516, 8.687154848939297440421022196735, 10.30865493469148430556997888510, 10.86378928719409194637113667134, 11.49025016979964945891749175245, 13.00367585485339188716691372910, 14.49703356369637803913610510678

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