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Properties

Label 2-276-12.11-c1-0-21
Degree $2$
Conductor $276$
Sign $0.186 + 0.982i$
Analytic cond. $2.20387$
Root an. cond. $1.48454$
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  + (−0.456 − 1.33i)2-s + (−0.784 + 1.54i)3-s + (−1.58 + 1.22i)4-s − 1.42i·5-s + (2.42 + 0.344i)6-s − 1.12i·7-s + (2.35 + 1.56i)8-s + (−1.77 − 2.42i)9-s + (−1.90 + 0.649i)10-s + 1.68·11-s + (−0.644 − 3.40i)12-s + 5.14·13-s + (−1.50 + 0.513i)14-s + (2.19 + 1.11i)15-s + (1.01 − 3.86i)16-s − 4.65i·17-s + ⋯
L(s)  = 1  + (−0.322 − 0.946i)2-s + (−0.452 + 0.891i)3-s + (−0.791 + 0.610i)4-s − 0.636i·5-s + (0.990 + 0.140i)6-s − 0.425i·7-s + (0.833 + 0.552i)8-s + (−0.590 − 0.807i)9-s + (−0.602 + 0.205i)10-s + 0.507·11-s + (−0.186 − 0.982i)12-s + 1.42·13-s + (−0.402 + 0.137i)14-s + (0.567 + 0.288i)15-s + (0.253 − 0.967i)16-s − 1.12i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.186 + 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.186 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(276\)    =    \(2^{2} \cdot 3 \cdot 23\)
Sign: $0.186 + 0.982i$
Analytic conductor: \(2.20387\)
Root analytic conductor: \(1.48454\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{276} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 276,\ (\ :1/2),\ 0.186 + 0.982i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.670730 - 0.555573i\)
\(L(\frac12)\) \(\approx\) \(0.670730 - 0.555573i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.456 + 1.33i)T \)
3 \( 1 + (0.784 - 1.54i)T \)
23 \( 1 + T \)
good5 \( 1 + 1.42iT - 5T^{2} \)
7 \( 1 + 1.12iT - 7T^{2} \)
11 \( 1 - 1.68T + 11T^{2} \)
13 \( 1 - 5.14T + 13T^{2} \)
17 \( 1 + 4.65iT - 17T^{2} \)
19 \( 1 + 5.86iT - 19T^{2} \)
29 \( 1 + 3.67iT - 29T^{2} \)
31 \( 1 - 3.79iT - 31T^{2} \)
37 \( 1 + 2.52T + 37T^{2} \)
41 \( 1 - 8.23iT - 41T^{2} \)
43 \( 1 - 3.72iT - 43T^{2} \)
47 \( 1 + 7.62T + 47T^{2} \)
53 \( 1 + 2.23iT - 53T^{2} \)
59 \( 1 - 3.98T + 59T^{2} \)
61 \( 1 + 7.14T + 61T^{2} \)
67 \( 1 + 12.2iT - 67T^{2} \)
71 \( 1 - 3.55T + 71T^{2} \)
73 \( 1 - 13.8T + 73T^{2} \)
79 \( 1 - 5.27iT - 79T^{2} \)
83 \( 1 + 12.9T + 83T^{2} \)
89 \( 1 - 15.9iT - 89T^{2} \)
97 \( 1 + 11.9T + 97T^{2} \)
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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

−11.34350445206089183174877826092, −10.96333789233621828474536473299, −9.766915677963044096224951281253, −9.113004763181452587531870810229, −8.308593055904028351485106790751, −6.66429475840716951450134743389, −5.11388310240615633486660335207, −4.30142340353530956935494967346, −3.16564770983114775380089081449, −0.910407994674419831763161235914, 1.56861565431745221342654222744, 3.80100728948319075312021287785, 5.60978640272925806483330102904, 6.20809269337055899067651647298, 7.04022628917522463414554127585, 8.171350959771423385621695986078, 8.801848594493686468854084718022, 10.30883211029700891249182057020, 11.00303118761748762216432691673, 12.21335900920693945157352271741

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