Properties

Label 2-177-177.113-c1-0-13
Degree $2$
Conductor $177$
Sign $0.840 + 0.542i$
Analytic cond. $1.41335$
Root an. cond. $1.18884$
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  + (1.21 + 0.410i)2-s + (0.669 − 1.59i)3-s + (−0.273 − 0.207i)4-s + (0.772 + 0.307i)5-s + (1.47 − 1.67i)6-s + (1.10 − 0.510i)7-s + (−1.69 − 2.49i)8-s + (−2.10 − 2.13i)9-s + (0.816 + 0.693i)10-s + (−1.30 + 4.68i)11-s + (−0.514 + 0.297i)12-s + (3.21 + 1.70i)13-s + (1.55 − 0.169i)14-s + (1.00 − 1.02i)15-s + (−0.854 − 3.07i)16-s + (−2.22 + 4.80i)17-s + ⋯
L(s)  = 1  + (0.862 + 0.290i)2-s + (0.386 − 0.922i)3-s + (−0.136 − 0.103i)4-s + (0.345 + 0.137i)5-s + (0.601 − 0.683i)6-s + (0.416 − 0.192i)7-s + (−0.598 − 0.882i)8-s + (−0.701 − 0.712i)9-s + (0.258 + 0.219i)10-s + (−0.392 + 1.41i)11-s + (−0.148 + 0.0858i)12-s + (0.892 + 0.472i)13-s + (0.415 − 0.0452i)14-s + (0.260 − 0.265i)15-s + (−0.213 − 0.769i)16-s + (−0.539 + 1.16i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.840 + 0.542i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.840 + 0.542i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.840 + 0.542i$
Analytic conductor: \(1.41335\)
Root analytic conductor: \(1.18884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :1/2),\ 0.840 + 0.542i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.75671 - 0.517453i\)
\(L(\frac12)\) \(\approx\) \(1.75671 - 0.517453i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.669 + 1.59i)T \)
59 \( 1 + (-7.62 + 0.964i)T \)
good2 \( 1 + (-1.21 - 0.410i)T + (1.59 + 1.21i)T^{2} \)
5 \( 1 + (-0.772 - 0.307i)T + (3.62 + 3.43i)T^{2} \)
7 \( 1 + (-1.10 + 0.510i)T + (4.53 - 5.33i)T^{2} \)
11 \( 1 + (1.30 - 4.68i)T + (-9.42 - 5.67i)T^{2} \)
13 \( 1 + (-3.21 - 1.70i)T + (7.29 + 10.7i)T^{2} \)
17 \( 1 + (2.22 - 4.80i)T + (-11.0 - 12.9i)T^{2} \)
19 \( 1 + (-3.68 + 0.810i)T + (17.2 - 7.97i)T^{2} \)
23 \( 1 + (-0.277 + 1.69i)T + (-21.7 - 7.34i)T^{2} \)
29 \( 1 + (-0.917 - 2.72i)T + (-23.0 + 17.5i)T^{2} \)
31 \( 1 + (1.15 - 5.23i)T + (-28.1 - 13.0i)T^{2} \)
37 \( 1 + (6.71 + 4.55i)T + (13.6 + 34.3i)T^{2} \)
41 \( 1 + (2.78 - 0.457i)T + (38.8 - 13.0i)T^{2} \)
43 \( 1 + (-6.12 + 1.70i)T + (36.8 - 22.1i)T^{2} \)
47 \( 1 + (3.65 + 9.17i)T + (-34.1 + 32.3i)T^{2} \)
53 \( 1 + (3.72 - 3.16i)T + (8.57 - 52.3i)T^{2} \)
61 \( 1 + (-3.00 + 8.90i)T + (-48.5 - 36.9i)T^{2} \)
67 \( 1 + (-7.24 + 4.91i)T + (24.7 - 62.2i)T^{2} \)
71 \( 1 + (-2.02 + 0.805i)T + (51.5 - 48.8i)T^{2} \)
73 \( 1 + (1.25 + 11.5i)T + (-71.2 + 15.6i)T^{2} \)
79 \( 1 + (-6.41 + 3.85i)T + (37.0 - 69.7i)T^{2} \)
83 \( 1 + (-0.120 - 2.23i)T + (-82.5 + 8.97i)T^{2} \)
89 \( 1 + (5.12 - 1.72i)T + (70.8 - 53.8i)T^{2} \)
97 \( 1 + (0.674 - 6.20i)T + (-94.7 - 20.8i)T^{2} \)
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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

−12.78270169690233909182349810450, −12.14327395998255611223166260242, −10.69547668767689151127424781904, −9.482588676561221208425604217993, −8.407902695318131881787575792116, −7.10182923445458665453893793258, −6.31095960270426619307509154952, −5.05929994068017075231362462706, −3.71928769838181407687291286664, −1.84624378087749094768536149585, 2.81210758294698042832759622785, 3.78085920632454482843468812001, 5.15105103300345524271254833048, 5.75876469517123112440133085111, 8.015097192216494952805437560293, 8.759382130200410764270976419750, 9.748943200710956340640252800848, 11.21233785251175349000290613297, 11.51560184179880485730220901511, 13.21772625279274159911700630593

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