Properties

Label 2-110-55.28-c1-0-5
Degree $2$
Conductor $110$
Sign $-0.511 + 0.859i$
Analytic cond. $0.878354$
Root an. cond. $0.937205$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.453 − 0.891i)2-s + (−0.443 − 2.79i)3-s + (−0.587 − 0.809i)4-s + (−1.71 + 1.44i)5-s + (−2.69 − 0.875i)6-s + (3.26 + 0.516i)7-s + (−0.987 + 0.156i)8-s + (−4.78 + 1.55i)9-s + (0.506 + 2.17i)10-s + (−0.266 − 3.30i)11-s + (−2.00 + 2.00i)12-s + (2.49 + 1.27i)13-s + (1.94 − 2.67i)14-s + (4.78 + 4.14i)15-s + (−0.309 + 0.951i)16-s + (2.57 − 1.30i)17-s + ⋯
L(s)  = 1  + (0.321 − 0.630i)2-s + (−0.255 − 1.61i)3-s + (−0.293 − 0.404i)4-s + (−0.764 + 0.644i)5-s + (−1.10 − 0.357i)6-s + (1.23 + 0.195i)7-s + (−0.349 + 0.0553i)8-s + (−1.59 + 0.518i)9-s + (0.160 + 0.688i)10-s + (−0.0803 − 0.996i)11-s + (−0.578 + 0.578i)12-s + (0.692 + 0.353i)13-s + (0.518 − 0.713i)14-s + (1.23 + 1.07i)15-s + (−0.0772 + 0.237i)16-s + (0.623 − 0.317i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.511 + 0.859i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.511 + 0.859i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(110\)    =    \(2 \cdot 5 \cdot 11\)
Sign: $-0.511 + 0.859i$
Analytic conductor: \(0.878354\)
Root analytic conductor: \(0.937205\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{110} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 110,\ (\ :1/2),\ -0.511 + 0.859i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.524206 - 0.921719i\)
\(L(\frac12)\) \(\approx\) \(0.524206 - 0.921719i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.453 + 0.891i)T \)
5 \( 1 + (1.71 - 1.44i)T \)
11 \( 1 + (0.266 + 3.30i)T \)
good3 \( 1 + (0.443 + 2.79i)T + (-2.85 + 0.927i)T^{2} \)
7 \( 1 + (-3.26 - 0.516i)T + (6.65 + 2.16i)T^{2} \)
13 \( 1 + (-2.49 - 1.27i)T + (7.64 + 10.5i)T^{2} \)
17 \( 1 + (-2.57 + 1.30i)T + (9.99 - 13.7i)T^{2} \)
19 \( 1 + (0.213 + 0.155i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-3.16 - 3.16i)T + 23iT^{2} \)
29 \( 1 + (6.91 - 5.02i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-2.80 - 8.63i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (0.251 - 1.58i)T + (-35.1 - 11.4i)T^{2} \)
41 \( 1 + (2.34 - 3.22i)T + (-12.6 - 38.9i)T^{2} \)
43 \( 1 + (-0.539 + 0.539i)T - 43iT^{2} \)
47 \( 1 + (6.91 - 1.09i)T + (44.6 - 14.5i)T^{2} \)
53 \( 1 + (-3.38 + 6.64i)T + (-31.1 - 42.8i)T^{2} \)
59 \( 1 + (-1.61 - 2.22i)T + (-18.2 + 56.1i)T^{2} \)
61 \( 1 + (2.35 + 0.763i)T + (49.3 + 35.8i)T^{2} \)
67 \( 1 + (1.62 - 1.62i)T - 67iT^{2} \)
71 \( 1 + (-3.37 + 10.3i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (0.297 - 1.88i)T + (-69.4 - 22.5i)T^{2} \)
79 \( 1 + (4.10 + 12.6i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (0.378 + 0.743i)T + (-48.7 + 67.1i)T^{2} \)
89 \( 1 - 17.3iT - 89T^{2} \)
97 \( 1 + (-4.57 - 2.33i)T + (57.0 + 78.4i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.26913057562506234560406131628, −12.08126674120990335626077442021, −11.44171087136919701511398331368, −10.86355172339269833962541292464, −8.640573086932676782608207995690, −7.77300071267821978158811759745, −6.63884852360635083871112304521, −5.28358160145063299063088656916, −3.23906436422842620729365106619, −1.44359799081257878435217138776, 3.92066349295055219683339004772, 4.62412727972157031088563004401, 5.57752346442920088817424986159, 7.64421311355892131989206797486, 8.548878049681955704748089922636, 9.716220868041262171436591406002, 10.94392507029352242649465611662, 11.75061611234279939851984906046, 13.03772969372729645762370247492, 14.56726201623800865979853669453

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