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Properties

Label 2-176-11.3-c3-0-13
Degree $2$
Conductor $176$
Sign $-0.510 + 0.859i$
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  + (−1.71 − 5.26i)3-s + (5.02 + 3.65i)5-s + (3.12 − 9.60i)7-s + (−2.98 + 2.17i)9-s + (17.8 − 31.8i)11-s + (−16.0 + 11.6i)13-s + (10.6 − 32.7i)15-s + (−0.0229 − 0.0166i)17-s + (−22.3 − 68.8i)19-s − 55.9·21-s − 92.3·23-s + (−26.6 − 82.1i)25-s + (−104. − 75.8i)27-s + (25.7 − 79.2i)29-s + (−139. + 101. i)31-s + ⋯
L(s)  = 1  + (−0.329 − 1.01i)3-s + (0.449 + 0.326i)5-s + (0.168 − 0.518i)7-s + (−0.110 + 0.0804i)9-s + (0.489 − 0.871i)11-s + (−0.342 + 0.248i)13-s + (0.183 − 0.563i)15-s + (−0.000327 − 0.000237i)17-s + (−0.270 − 0.831i)19-s − 0.581·21-s − 0.837·23-s + (−0.213 − 0.656i)25-s + (−0.744 − 0.540i)27-s + (0.164 − 0.507i)29-s + (−0.806 + 0.585i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.510 + 0.859i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.510 + 0.859i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(176\)    =    \(2^{4} \cdot 11\)
Sign: $-0.510 + 0.859i$
Analytic conductor: \(10.3843\)
Root analytic conductor: \(3.22247\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{176} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 176,\ (\ :3/2),\ -0.510 + 0.859i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.718390 - 1.26196i\)
\(L(\frac12)\) \(\approx\) \(0.718390 - 1.26196i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-17.8 + 31.8i)T \)
good3 \( 1 + (1.71 + 5.26i)T + (-21.8 + 15.8i)T^{2} \)
5 \( 1 + (-5.02 - 3.65i)T + (38.6 + 118. i)T^{2} \)
7 \( 1 + (-3.12 + 9.60i)T + (-277. - 201. i)T^{2} \)
13 \( 1 + (16.0 - 11.6i)T + (678. - 2.08e3i)T^{2} \)
17 \( 1 + (0.0229 + 0.0166i)T + (1.51e3 + 4.67e3i)T^{2} \)
19 \( 1 + (22.3 + 68.8i)T + (-5.54e3 + 4.03e3i)T^{2} \)
23 \( 1 + 92.3T + 1.21e4T^{2} \)
29 \( 1 + (-25.7 + 79.2i)T + (-1.97e4 - 1.43e4i)T^{2} \)
31 \( 1 + (139. - 101. i)T + (9.20e3 - 2.83e4i)T^{2} \)
37 \( 1 + (-55.1 + 169. i)T + (-4.09e4 - 2.97e4i)T^{2} \)
41 \( 1 + (72.8 + 224. i)T + (-5.57e4 + 4.05e4i)T^{2} \)
43 \( 1 - 69.3T + 7.95e4T^{2} \)
47 \( 1 + (-133. - 411. i)T + (-8.39e4 + 6.10e4i)T^{2} \)
53 \( 1 + (-485. + 352. i)T + (4.60e4 - 1.41e5i)T^{2} \)
59 \( 1 + (-112. + 347. i)T + (-1.66e5 - 1.20e5i)T^{2} \)
61 \( 1 + (-554. - 402. i)T + (7.01e4 + 2.15e5i)T^{2} \)
67 \( 1 + 224.T + 3.00e5T^{2} \)
71 \( 1 + (206. + 149. i)T + (1.10e5 + 3.40e5i)T^{2} \)
73 \( 1 + (272. - 839. i)T + (-3.14e5 - 2.28e5i)T^{2} \)
79 \( 1 + (-313. + 227. i)T + (1.52e5 - 4.68e5i)T^{2} \)
83 \( 1 + (-982. - 713. i)T + (1.76e5 + 5.43e5i)T^{2} \)
89 \( 1 + 1.23e3T + 7.04e5T^{2} \)
97 \( 1 + (-721. + 524. i)T + (2.82e5 - 8.68e5i)T^{2} \)
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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

−11.95214368891052298800485611507, −11.07525202700419260357191148339, −10.00698772719132950077704480427, −8.772916946961071814739325647107, −7.50428164913648147444546606800, −6.66818872195130622200263428060, −5.76753020402664072993714902218, −4.04109550715741824660780099943, −2.21300133449167848400267559672, −0.69447831537123836678550334161, 1.91119748367182457060121704679, 3.88462508918545446910209712386, 4.98112665475190218367713209245, 5.89125544918044216857037763910, 7.41976672922817227392056134190, 8.798483233422141972924651327242, 9.754830634405246086204343557557, 10.32631821530014497362912514286, 11.58814727240615252367444460103, 12.43439829855059105569110124964

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