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Properties

Label 2-176-11.9-c3-0-4
Degree $2$
Conductor $176$
Sign $-0.962 - 0.271i$
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  + (−6.31 + 4.59i)3-s + (4.60 + 14.1i)5-s + (17.6 + 12.7i)7-s + (10.5 − 32.3i)9-s + (29.3 + 21.6i)11-s + (−13.6 + 41.8i)13-s + (−94.2 − 68.4i)15-s + (−7.69 − 23.6i)17-s + (17.7 − 12.9i)19-s − 170.·21-s − 177.·23-s + (−78.9 + 57.3i)25-s + (16.9 + 52.1i)27-s + (120. + 87.8i)29-s + (−23.2 + 71.4i)31-s + ⋯
L(s)  = 1  + (−1.21 + 0.883i)3-s + (0.412 + 1.26i)5-s + (0.950 + 0.690i)7-s + (0.389 − 1.19i)9-s + (0.805 + 0.592i)11-s + (−0.290 + 0.893i)13-s + (−1.62 − 1.17i)15-s + (−0.109 − 0.338i)17-s + (0.214 − 0.155i)19-s − 1.76·21-s − 1.61·23-s + (−0.631 + 0.458i)25-s + (0.120 + 0.371i)27-s + (0.773 + 0.562i)29-s + (−0.134 + 0.414i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.962 - 0.271i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.962 - 0.271i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(176\)    =    \(2^{4} \cdot 11\)
Sign: $-0.962 - 0.271i$
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: \((2,\ 176,\ (\ :3/2),\ -0.962 - 0.271i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.158488 + 1.14607i\)
\(L(\frac12)\) \(\approx\) \(0.158488 + 1.14607i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-29.3 - 21.6i)T \)
good3 \( 1 + (6.31 - 4.59i)T + (8.34 - 25.6i)T^{2} \)
5 \( 1 + (-4.60 - 14.1i)T + (-101. + 73.4i)T^{2} \)
7 \( 1 + (-17.6 - 12.7i)T + (105. + 326. i)T^{2} \)
13 \( 1 + (13.6 - 41.8i)T + (-1.77e3 - 1.29e3i)T^{2} \)
17 \( 1 + (7.69 + 23.6i)T + (-3.97e3 + 2.88e3i)T^{2} \)
19 \( 1 + (-17.7 + 12.9i)T + (2.11e3 - 6.52e3i)T^{2} \)
23 \( 1 + 177.T + 1.21e4T^{2} \)
29 \( 1 + (-120. - 87.8i)T + (7.53e3 + 2.31e4i)T^{2} \)
31 \( 1 + (23.2 - 71.4i)T + (-2.41e4 - 1.75e4i)T^{2} \)
37 \( 1 + (179. + 130. i)T + (1.56e4 + 4.81e4i)T^{2} \)
41 \( 1 + (-204. + 148. i)T + (2.12e4 - 6.55e4i)T^{2} \)
43 \( 1 + 130.T + 7.95e4T^{2} \)
47 \( 1 + (-403. + 293. i)T + (3.20e4 - 9.87e4i)T^{2} \)
53 \( 1 + (-3.99 + 12.3i)T + (-1.20e5 - 8.75e4i)T^{2} \)
59 \( 1 + (-28.7 - 20.9i)T + (6.34e4 + 1.95e5i)T^{2} \)
61 \( 1 + (-166. - 511. i)T + (-1.83e5 + 1.33e5i)T^{2} \)
67 \( 1 - 519.T + 3.00e5T^{2} \)
71 \( 1 + (24.2 + 74.6i)T + (-2.89e5 + 2.10e5i)T^{2} \)
73 \( 1 + (925. + 672. i)T + (1.20e5 + 3.69e5i)T^{2} \)
79 \( 1 + (238. - 734. i)T + (-3.98e5 - 2.89e5i)T^{2} \)
83 \( 1 + (166. + 510. i)T + (-4.62e5 + 3.36e5i)T^{2} \)
89 \( 1 - 667.T + 7.04e5T^{2} \)
97 \( 1 + (55.5 - 170. i)T + (-7.38e5 - 5.36e5i)T^{2} \)
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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

−12.01055430916688127505426785519, −11.74109008606248220879551091436, −10.70475099813158411072491343560, −10.02348037195555023939484251857, −8.938378903283062997109662719134, −7.17409428840048375787364331531, −6.23653353291206093584729313509, −5.19258247565194773246574475904, −4.07605791012163892716482239740, −2.12167124782638122110487723805, 0.65506171105411039948250730180, 1.53926379329187845730740330808, 4.34091561656575601612885957250, 5.44016531458123372934731147450, 6.24196000001617629658914938300, 7.64933377623395997943579133252, 8.483074969124330088215387137369, 9.945637492853117345658490895189, 11.06937011837166906158336516504, 11.92462165374188640310646527492

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