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Properties

Label 2-275-11.3-c1-0-3
Degree $2$
Conductor $275$
Sign $-0.941 - 0.335i$
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  + (−0.596 + 0.433i)2-s + (0.868 + 2.67i)3-s + (−0.449 + 1.38i)4-s + (−1.67 − 1.21i)6-s + (−0.318 + 0.980i)7-s + (−0.787 − 2.42i)8-s + (−3.96 + 2.88i)9-s + (1.93 + 2.69i)11-s − 4.09·12-s + (2.79 − 2.02i)13-s + (−0.235 − 0.723i)14-s + (−0.834 − 0.606i)16-s + (1.94 + 1.40i)17-s + (1.11 − 3.44i)18-s + (−2.36 − 7.29i)19-s + ⋯
L(s)  = 1  + (−0.421 + 0.306i)2-s + (0.501 + 1.54i)3-s + (−0.224 + 0.692i)4-s + (−0.684 − 0.497i)6-s + (−0.120 + 0.370i)7-s + (−0.278 − 0.857i)8-s + (−1.32 + 0.961i)9-s + (0.583 + 0.811i)11-s − 1.18·12-s + (0.773 − 0.562i)13-s + (−0.0628 − 0.193i)14-s + (−0.208 − 0.151i)16-s + (0.470 + 0.341i)17-s + (0.263 − 0.811i)18-s + (−0.543 − 1.67i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.941 - 0.335i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.941 - 0.335i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(275\)    =    \(5^{2} \cdot 11\)
Sign: $-0.941 - 0.335i$
Analytic conductor: \(2.19588\)
Root analytic conductor: \(1.48185\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{275} (201, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 275,\ (\ :1/2),\ -0.941 - 0.335i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.184054 + 1.06372i\)
\(L(\frac12)\) \(\approx\) \(0.184054 + 1.06372i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.93 - 2.69i)T \)
good2 \( 1 + (0.596 - 0.433i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 + (-0.868 - 2.67i)T + (-2.42 + 1.76i)T^{2} \)
7 \( 1 + (0.318 - 0.980i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-2.79 + 2.02i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-1.94 - 1.40i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (2.36 + 7.29i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 2.45T + 23T^{2} \)
29 \( 1 + (1.83 - 5.66i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-2.98 + 2.16i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (1.84 - 5.66i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-1.21 - 3.74i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 7.64T + 43T^{2} \)
47 \( 1 + (1.80 + 5.55i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (9.58 - 6.96i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-0.910 + 2.80i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (2.00 + 1.45i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 6.14T + 67T^{2} \)
71 \( 1 + (1.63 + 1.18i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-0.255 + 0.785i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-9.77 + 7.09i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-1.30 - 0.946i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 8.16T + 89T^{2} \)
97 \( 1 + (-1.97 + 1.43i)T + (29.9 - 92.2i)T^{2} \)
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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

−12.29256209987742689583671515435, −11.11483130954298291332237000572, −10.14792791172675280165156760288, −9.246491536242774460199371178760, −8.802126412713698312571194093374, −7.78616205265615217417608876208, −6.43508092021702760568146445487, −4.87341144665704538857633196797, −3.95182034230035503009936128645, −2.94130066996149004232867042177, 0.975830049211209716124609238278, 2.07654363690590482183104590860, 3.79222470031274135295405839594, 5.85716522758775307815197073030, 6.44018370702073265930237664388, 7.77033148578557368954739801480, 8.504240718663721882443289200845, 9.410162570097340852928621992647, 10.56273915047325488227954808302, 11.55608969893853194674985720476

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