Properties

Label 2-55-55.7-c1-0-3
Degree $2$
Conductor $55$
Sign $0.962 + 0.272i$
Analytic cond. $0.439177$
Root an. cond. $0.662704$
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  + (1.50 − 0.237i)2-s + (−0.361 − 0.710i)3-s + (0.295 − 0.0959i)4-s + (−1.89 + 1.18i)5-s + (−0.712 − 0.980i)6-s + (0.170 + 0.0869i)7-s + (−2.28 + 1.16i)8-s + (1.38 − 1.91i)9-s + (−2.56 + 2.23i)10-s + (1.77 + 2.80i)11-s + (−0.175 − 0.175i)12-s + (−0.484 − 3.05i)13-s + (0.276 + 0.0899i)14-s + (1.52 + 0.916i)15-s + (−3.66 + 2.65i)16-s + (0.579 − 3.66i)17-s + ⋯
L(s)  = 1  + (1.06 − 0.168i)2-s + (−0.208 − 0.410i)3-s + (0.147 − 0.0479i)4-s + (−0.847 + 0.531i)5-s + (−0.290 − 0.400i)6-s + (0.0644 + 0.0328i)7-s + (−0.808 + 0.412i)8-s + (0.463 − 0.637i)9-s + (−0.810 + 0.706i)10-s + (0.535 + 0.844i)11-s + (−0.0505 − 0.0505i)12-s + (−0.134 − 0.847i)13-s + (0.0739 + 0.0240i)14-s + (0.394 + 0.236i)15-s + (−0.915 + 0.664i)16-s + (0.140 − 0.887i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 + 0.272i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.962 + 0.272i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(55\)    =    \(5 \cdot 11\)
Sign: $0.962 + 0.272i$
Analytic conductor: \(0.439177\)
Root analytic conductor: \(0.662704\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{55} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 55,\ (\ :1/2),\ 0.962 + 0.272i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.09576 - 0.151985i\)
\(L(\frac12)\) \(\approx\) \(1.09576 - 0.151985i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.89 - 1.18i)T \)
11 \( 1 + (-1.77 - 2.80i)T \)
good2 \( 1 + (-1.50 + 0.237i)T + (1.90 - 0.618i)T^{2} \)
3 \( 1 + (0.361 + 0.710i)T + (-1.76 + 2.42i)T^{2} \)
7 \( 1 + (-0.170 - 0.0869i)T + (4.11 + 5.66i)T^{2} \)
13 \( 1 + (0.484 + 3.05i)T + (-12.3 + 4.01i)T^{2} \)
17 \( 1 + (-0.579 + 3.66i)T + (-16.1 - 5.25i)T^{2} \)
19 \( 1 + (0.229 - 0.707i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (1.14 - 1.14i)T - 23iT^{2} \)
29 \( 1 + (-2.95 - 9.07i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (0.283 + 0.206i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-2.45 + 4.81i)T + (-21.7 - 29.9i)T^{2} \)
41 \( 1 + (-6.36 - 2.06i)T + (33.1 + 24.0i)T^{2} \)
43 \( 1 + (-3.72 - 3.72i)T + 43iT^{2} \)
47 \( 1 + (11.0 - 5.61i)T + (27.6 - 38.0i)T^{2} \)
53 \( 1 + (8.91 - 1.41i)T + (50.4 - 16.3i)T^{2} \)
59 \( 1 + (9.15 - 2.97i)T + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (3.46 + 4.76i)T + (-18.8 + 58.0i)T^{2} \)
67 \( 1 + (-4.13 - 4.13i)T + 67iT^{2} \)
71 \( 1 + (-9.27 + 6.73i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-1.09 + 2.14i)T + (-42.9 - 59.0i)T^{2} \)
79 \( 1 + (0.542 + 0.394i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-16.4 - 2.60i)T + (78.9 + 25.6i)T^{2} \)
89 \( 1 + 7.92iT - 89T^{2} \)
97 \( 1 + (0.215 + 1.36i)T + (-92.2 + 29.9i)T^{2} \)
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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

−14.94948949134718142341534156994, −14.27779127748408900988176634829, −12.72571868475217883385826032549, −12.26552973312307471372233948111, −11.16455561525279091905851499164, −9.456767161591735597368879100919, −7.68643550917660765081057845424, −6.44469236181763174919793039982, −4.70628302067454926145386025398, −3.31015956006960652487594756453, 3.88747454656464673825463355277, 4.76372173238952839570985412011, 6.28821022215102590594472111980, 8.104329750597607940108101225773, 9.481372192115276486374264572080, 11.13620323631857164829428137245, 12.13560995938416295064774464248, 13.23061348631424941008031586843, 14.22382895506289199281155610618, 15.36074669824424416083226904738

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