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Properties

Label 2-176-11.9-c3-0-6
Degree $2$
Conductor $176$
Sign $0.682 - 0.730i$
Analytic cond. $10.3843$
Root an. cond. $3.22247$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.87 − 1.36i)3-s + (1.88 + 5.80i)5-s + (3.69 + 2.68i)7-s + (−6.68 + 20.5i)9-s + (32.6 − 16.1i)11-s + (−14.6 + 45.1i)13-s + (11.4 + 8.31i)15-s + (19.6 + 60.5i)17-s + (14.9 − 10.8i)19-s + 10.5·21-s + 51.9·23-s + (71.0 − 51.5i)25-s + (34.8 + 107. i)27-s + (62.4 + 45.3i)29-s + (−33.7 + 103. i)31-s + ⋯
L(s)  = 1  + (0.360 − 0.262i)3-s + (0.168 + 0.519i)5-s + (0.199 + 0.145i)7-s + (−0.247 + 0.761i)9-s + (0.896 − 0.443i)11-s + (−0.313 + 0.963i)13-s + (0.196 + 0.143i)15-s + (0.280 + 0.863i)17-s + (0.180 − 0.130i)19-s + 0.110·21-s + 0.470·23-s + (0.568 − 0.412i)25-s + (0.248 + 0.764i)27-s + (0.399 + 0.290i)29-s + (−0.195 + 0.601i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.682 - 0.730i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.682 - 0.730i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(176\)    =    \(2^{4} \cdot 11\)
Sign: $0.682 - 0.730i$
Analytic conductor: \(10.3843\)
Root analytic conductor: \(3.22247\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{176} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 176,\ (\ :3/2),\ 0.682 - 0.730i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.82204 + 0.791430i\)
\(L(\frac12)\) \(\approx\) \(1.82204 + 0.791430i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 + (-32.6 + 16.1i)T \)
good3 \( 1 + (-1.87 + 1.36i)T + (8.34 - 25.6i)T^{2} \)
5 \( 1 + (-1.88 - 5.80i)T + (-101. + 73.4i)T^{2} \)
7 \( 1 + (-3.69 - 2.68i)T + (105. + 326. i)T^{2} \)
13 \( 1 + (14.6 - 45.1i)T + (-1.77e3 - 1.29e3i)T^{2} \)
17 \( 1 + (-19.6 - 60.5i)T + (-3.97e3 + 2.88e3i)T^{2} \)
19 \( 1 + (-14.9 + 10.8i)T + (2.11e3 - 6.52e3i)T^{2} \)
23 \( 1 - 51.9T + 1.21e4T^{2} \)
29 \( 1 + (-62.4 - 45.3i)T + (7.53e3 + 2.31e4i)T^{2} \)
31 \( 1 + (33.7 - 103. i)T + (-2.41e4 - 1.75e4i)T^{2} \)
37 \( 1 + (-113. - 82.2i)T + (1.56e4 + 4.81e4i)T^{2} \)
41 \( 1 + (-128. + 93.0i)T + (2.12e4 - 6.55e4i)T^{2} \)
43 \( 1 - 26.9T + 7.95e4T^{2} \)
47 \( 1 + (208. - 151. i)T + (3.20e4 - 9.87e4i)T^{2} \)
53 \( 1 + (45.7 - 140. i)T + (-1.20e5 - 8.75e4i)T^{2} \)
59 \( 1 + (587. + 426. i)T + (6.34e4 + 1.95e5i)T^{2} \)
61 \( 1 + (-149. - 458. i)T + (-1.83e5 + 1.33e5i)T^{2} \)
67 \( 1 - 464.T + 3.00e5T^{2} \)
71 \( 1 + (281. + 867. i)T + (-2.89e5 + 2.10e5i)T^{2} \)
73 \( 1 + (742. + 539. i)T + (1.20e5 + 3.69e5i)T^{2} \)
79 \( 1 + (-234. + 722. i)T + (-3.98e5 - 2.89e5i)T^{2} \)
83 \( 1 + (215. + 664. i)T + (-4.62e5 + 3.36e5i)T^{2} \)
89 \( 1 + 564.T + 7.04e5T^{2} \)
97 \( 1 + (-360. + 1.11e3i)T + (-7.38e5 - 5.36e5i)T^{2} \)
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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

−12.35397087613498507984358734366, −11.33419348465688186657240698350, −10.50419134556927447737178319959, −9.205589792765675188769053533762, −8.338870482694048892657245020225, −7.14837100983018846149094610746, −6.15692132562634857711152967656, −4.67604572705329734154491644087, −3.11211683491861607113862669799, −1.69678329929112912365095075796, 0.961644271724443365596378499275, 2.95808703344081750703089418476, 4.32428506584468720062616205222, 5.56153673108750665082171600423, 6.91664051053563523630170045760, 8.126786360255282345864972903021, 9.249757276154390021493801610967, 9.795640622736835750053998431136, 11.20587417205177800834871885680, 12.18341077881900876170253473681

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