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Properties

Label 2-275-55.14-c1-0-15
Degree $2$
Conductor $275$
Sign $-0.966 + 0.257i$
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  + (0.0549 + 0.0756i)2-s + (−1.39 + 0.453i)3-s + (0.615 − 1.89i)4-s + (−0.110 − 0.0806i)6-s + (−4.30 − 1.39i)7-s + (0.354 − 0.115i)8-s + (−0.686 + 0.498i)9-s + (−2.39 + 2.29i)11-s + 2.92i·12-s + (0.671 + 0.924i)13-s + (−0.130 − 0.402i)14-s + (−3.19 − 2.32i)16-s + (−1.98 + 2.72i)17-s + (−0.0754 − 0.0245i)18-s + (−1.88 − 5.78i)19-s + ⋯
L(s)  = 1  + (0.0388 + 0.0534i)2-s + (−0.805 + 0.261i)3-s + (0.307 − 0.946i)4-s + (−0.0452 − 0.0329i)6-s + (−1.62 − 0.528i)7-s + (0.125 − 0.0407i)8-s + (−0.228 + 0.166i)9-s + (−0.723 + 0.690i)11-s + 0.843i·12-s + (0.186 + 0.256i)13-s + (−0.0349 − 0.107i)14-s + (−0.798 − 0.580i)16-s + (−0.480 + 0.661i)17-s + (−0.0177 − 0.00578i)18-s + (−0.431 − 1.32i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.966 + 0.257i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.966 + 0.257i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(275\)    =    \(5^{2} \cdot 11\)
Sign: $-0.966 + 0.257i$
Analytic conductor: \(2.19588\)
Root analytic conductor: \(1.48185\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{275} (124, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 275,\ (\ :1/2),\ -0.966 + 0.257i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0298010 - 0.227986i\)
\(L(\frac12)\) \(\approx\) \(0.0298010 - 0.227986i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.39 - 2.29i)T \)
good2 \( 1 + (-0.0549 - 0.0756i)T + (-0.618 + 1.90i)T^{2} \)
3 \( 1 + (1.39 - 0.453i)T + (2.42 - 1.76i)T^{2} \)
7 \( 1 + (4.30 + 1.39i)T + (5.66 + 4.11i)T^{2} \)
13 \( 1 + (-0.671 - 0.924i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (1.98 - 2.72i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (1.88 + 5.78i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 5.45iT - 23T^{2} \)
29 \( 1 + (1.02 - 3.15i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (1.44 - 1.05i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (1.41 + 0.460i)T + (29.9 + 21.7i)T^{2} \)
41 \( 1 + (0.539 + 1.66i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 0.263iT - 43T^{2} \)
47 \( 1 + (-6.58 + 2.13i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (-0.846 - 1.16i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (-2.18 + 6.72i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (2.02 + 1.47i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 0.516iT - 67T^{2} \)
71 \( 1 + (-8.68 - 6.30i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (5.40 + 1.75i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (9.14 - 6.64i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-2.63 + 3.62i)T + (-25.6 - 78.9i)T^{2} \)
89 \( 1 + 13.2T + 89T^{2} \)
97 \( 1 + (1.97 + 2.71i)T + (-29.9 + 92.2i)T^{2} \)
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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

−11.10004974583938154155653486036, −10.56675660983160637644327634562, −9.897892387195400632612193415121, −8.822895670020548961392508453614, −7.00683524923726499527090824557, −6.47876519129978094022227840672, −5.45231125049240273762818696216, −4.36536504308364136550519815142, −2.55266575525756824241947465715, −0.17325257000099283544233676068, 2.75470945024111614007569228135, 3.67590832613783927586953437989, 5.62824612029426325305191009919, 6.25885458033551108026421142300, 7.30498563755726089846434139330, 8.461796284105127592643923498531, 9.456984371881163237940998462522, 10.64568520284813936313829228268, 11.60351940965737374483047968904, 12.28269794428883424427706043510

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