Properties

Label 2-275-11.5-c1-0-10
Degree $2$
Conductor $275$
Sign $0.778 + 0.627i$
Analytic cond. $2.19588$
Root an. cond. $1.48185$
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  + (−0.248 − 0.763i)2-s + (1.42 + 1.03i)3-s + (1.09 − 0.796i)4-s + (0.436 − 1.34i)6-s + (0.479 − 0.348i)7-s + (−2.17 − 1.58i)8-s + (0.0309 + 0.0953i)9-s + (1.96 − 2.67i)11-s + 2.38·12-s + (0.554 + 1.70i)13-s + (−0.384 − 0.279i)14-s + (0.169 − 0.522i)16-s + (−2.18 + 6.73i)17-s + (0.0650 − 0.0472i)18-s + (1.85 + 1.34i)19-s + ⋯
L(s)  = 1  + (−0.175 − 0.539i)2-s + (0.822 + 0.597i)3-s + (0.548 − 0.398i)4-s + (0.178 − 0.548i)6-s + (0.181 − 0.131i)7-s + (−0.770 − 0.559i)8-s + (0.0103 + 0.0317i)9-s + (0.591 − 0.806i)11-s + 0.689·12-s + (0.153 + 0.473i)13-s + (−0.102 − 0.0746i)14-s + (0.0424 − 0.130i)16-s + (−0.530 + 1.63i)17-s + (0.0153 − 0.0111i)18-s + (0.424 + 0.308i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.778 + 0.627i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.778 + 0.627i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(275\)    =    \(5^{2} \cdot 11\)
Sign: $0.778 + 0.627i$
Analytic conductor: \(2.19588\)
Root analytic conductor: \(1.48185\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{275} (126, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 275,\ (\ :1/2),\ 0.778 + 0.627i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.59388 - 0.562746i\)
\(L(\frac12)\) \(\approx\) \(1.59388 - 0.562746i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 + (-1.96 + 2.67i)T \)
good2 \( 1 + (0.248 + 0.763i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (-1.42 - 1.03i)T + (0.927 + 2.85i)T^{2} \)
7 \( 1 + (-0.479 + 0.348i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (-0.554 - 1.70i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (2.18 - 6.73i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-1.85 - 1.34i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 1.49T + 23T^{2} \)
29 \( 1 + (2.89 - 2.10i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-1.90 - 5.86i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-5.93 + 4.31i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (6.80 + 4.94i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 9.51T + 43T^{2} \)
47 \( 1 + (-1.56 - 1.13i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (0.736 + 2.26i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-0.0309 + 0.0224i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (1.06 - 3.27i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 6.79T + 67T^{2} \)
71 \( 1 + (3.64 - 11.2i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-5.51 + 4.01i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-1.39 - 4.30i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (1.83 - 5.65i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 6.21T + 89T^{2} \)
97 \( 1 + (-1.66 - 5.11i)T + (-78.4 + 57.0i)T^{2} \)
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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

−11.58261305484968328830398863118, −10.77299874748249817753283030344, −9.951656235362254688638573446433, −9.022552900224352929966938560648, −8.317581116124171002567527740484, −6.75955844277956509875054846006, −5.84611068703189419256304304840, −4.05327061725261882488491588951, −3.18438056961910313350905818820, −1.64459414499943548340160419212, 2.08433902639670553709802608960, 3.13662493068839428078386685327, 4.91706566122406160903851852386, 6.39829702351321821233938869334, 7.30356997056602051234700938921, 7.917344344276469546432691022885, 8.854058863226973992541690665396, 9.806127674138152852716671621034, 11.39044769451385743238883500649, 11.87537427368957630655818910996

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