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Properties

Label 2-275-55.19-c2-0-8
Degree $2$
Conductor $275$
Sign $-0.841 - 0.539i$
Analytic cond. $7.49320$
Root an. cond. $2.73737$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.48 + 1.80i)2-s + (−3.44 + 1.11i)3-s + (1.69 − 5.20i)4-s + (6.54 − 9.00i)6-s + (−0.277 + 0.854i)7-s + (1.40 + 4.30i)8-s + (3.30 − 2.40i)9-s + (10.8 − 1.67i)11-s + 19.7i·12-s + (6.88 − 5i)13-s + (−0.854 − 2.62i)14-s + (6.42 + 4.66i)16-s + (20.0 + 14.5i)17-s + (−3.88 + 11.9i)18-s + (−11.2 + 3.66i)19-s + ⋯
L(s)  = 1  + (−1.24 + 0.904i)2-s + (−1.14 + 0.372i)3-s + (0.422 − 1.30i)4-s + (1.09 − 1.50i)6-s + (−0.0396 + 0.122i)7-s + (0.175 + 0.538i)8-s + (0.367 − 0.267i)9-s + (0.988 − 0.152i)11-s + 1.64i·12-s + (0.529 − 0.384i)13-s + (−0.0610 − 0.187i)14-s + (0.401 + 0.291i)16-s + (1.17 + 0.854i)17-s + (−0.216 + 0.665i)18-s + (−0.593 + 0.192i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.841 - 0.539i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.841 - 0.539i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(275\)    =    \(5^{2} \cdot 11\)
Sign: $-0.841 - 0.539i$
Analytic conductor: \(7.49320\)
Root analytic conductor: \(2.73737\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{275} (74, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 275,\ (\ :1),\ -0.841 - 0.539i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.126415 + 0.431209i\)
\(L(\frac12)\) \(\approx\) \(0.126415 + 0.431209i\)
\(L(2)\) 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 \( 1 + (-10.8 + 1.67i)T \)
good2 \( 1 + (2.48 - 1.80i)T + (1.23 - 3.80i)T^{2} \)
3 \( 1 + (3.44 - 1.11i)T + (7.28 - 5.29i)T^{2} \)
7 \( 1 + (0.277 - 0.854i)T + (-39.6 - 28.8i)T^{2} \)
13 \( 1 + (-6.88 + 5i)T + (52.2 - 160. i)T^{2} \)
17 \( 1 + (-20.0 - 14.5i)T + (89.3 + 274. i)T^{2} \)
19 \( 1 + (11.2 - 3.66i)T + (292. - 212. i)T^{2} \)
23 \( 1 + 7.23iT - 529T^{2} \)
29 \( 1 + (-3.29 - 1.06i)T + (680. + 494. i)T^{2} \)
31 \( 1 + (26.7 - 19.4i)T + (296. - 913. i)T^{2} \)
37 \( 1 + (38.2 + 12.4i)T + (1.10e3 + 804. i)T^{2} \)
41 \( 1 + (-1.24 + 0.403i)T + (1.35e3 - 988. i)T^{2} \)
43 \( 1 + 33.0T + 1.84e3T^{2} \)
47 \( 1 + (-21.6 + 7.03i)T + (1.78e3 - 1.29e3i)T^{2} \)
53 \( 1 + (-46.1 - 63.5i)T + (-868. + 2.67e3i)T^{2} \)
59 \( 1 + (9.66 - 29.7i)T + (-2.81e3 - 2.04e3i)T^{2} \)
61 \( 1 + (16.5 - 22.7i)T + (-1.14e3 - 3.53e3i)T^{2} \)
67 \( 1 - 76.5iT - 4.48e3T^{2} \)
71 \( 1 + (-50.4 - 36.6i)T + (1.55e3 + 4.79e3i)T^{2} \)
73 \( 1 + (29.0 - 89.5i)T + (-4.31e3 - 3.13e3i)T^{2} \)
79 \( 1 + (38.7 + 53.3i)T + (-1.92e3 + 5.93e3i)T^{2} \)
83 \( 1 + (-45.8 - 33.3i)T + (2.12e3 + 6.55e3i)T^{2} \)
89 \( 1 - 62.2T + 7.92e3T^{2} \)
97 \( 1 + (-42.5 - 58.5i)T + (-2.90e3 + 8.94e3i)T^{2} \)
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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

−11.85235176771451313954316192748, −10.64215983216436795407183059698, −10.28224783307732222430582602640, −9.034294896760078705469966219935, −8.322822385293203408678110285044, −7.10640621467889124074994078639, −6.12879364875718991752029739842, −5.54159214554035452749700536932, −3.85734056846678067839958441656, −1.12399097901821645660144256158, 0.50070333422983255951401131889, 1.69502472930313252168522148786, 3.51545413130419865510961976275, 5.26799136427469271051420213372, 6.46213567008980992830762679758, 7.43923881475008528000531393738, 8.696895126754231966583789782445, 9.512835551165183769665640387528, 10.45498377243039056801343701016, 11.32076494273330127534624944383

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