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Properties

Label 2-275-11.4-c1-0-15
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: $\chi_{275} (26, \cdot )$
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

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

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