Properties

Label 2-177-177.101-c1-0-0
Degree $2$
Conductor $177$
Sign $-0.527 - 0.849i$
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  + (−1.27 + 1.88i)2-s + (−0.951 − 1.44i)3-s + (−1.16 − 2.93i)4-s + (0.346 − 0.749i)5-s + (3.93 + 0.0547i)6-s + (0.990 + 3.56i)7-s + (2.57 + 0.566i)8-s + (−1.18 + 2.75i)9-s + (0.967 + 1.60i)10-s + (1.13 + 1.07i)11-s + (−3.13 + 4.48i)12-s + (−0.644 + 5.93i)13-s + (−7.97 − 2.68i)14-s + (−1.41 + 0.211i)15-s + (0.240 − 0.228i)16-s + (2.93 + 0.815i)17-s + ⋯
L(s)  = 1  + (−0.901 + 1.32i)2-s + (−0.549 − 0.835i)3-s + (−0.584 − 1.46i)4-s + (0.155 − 0.335i)5-s + (1.60 + 0.0223i)6-s + (0.374 + 1.34i)7-s + (0.910 + 0.200i)8-s + (−0.395 + 0.918i)9-s + (0.305 + 0.508i)10-s + (0.343 + 0.325i)11-s + (−0.904 + 1.29i)12-s + (−0.178 + 1.64i)13-s + (−2.13 − 0.717i)14-s + (−0.365 + 0.0546i)15-s + (0.0602 − 0.0570i)16-s + (0.712 + 0.197i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.527 - 0.849i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.527 - 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-0.527 - 0.849i$
Analytic conductor: \(1.41335\)
Root analytic conductor: \(1.18884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :1/2),\ -0.527 - 0.849i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.276127 + 0.496411i\)
\(L(\frac12)\) \(\approx\) \(0.276127 + 0.496411i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.951 + 1.44i)T \)
59 \( 1 + (4.93 + 5.88i)T \)
good2 \( 1 + (1.27 - 1.88i)T + (-0.740 - 1.85i)T^{2} \)
5 \( 1 + (-0.346 + 0.749i)T + (-3.23 - 3.81i)T^{2} \)
7 \( 1 + (-0.990 - 3.56i)T + (-5.99 + 3.60i)T^{2} \)
11 \( 1 + (-1.13 - 1.07i)T + (0.595 + 10.9i)T^{2} \)
13 \( 1 + (0.644 - 5.93i)T + (-12.6 - 2.79i)T^{2} \)
17 \( 1 + (-2.93 - 0.815i)T + (14.5 + 8.76i)T^{2} \)
19 \( 1 + (2.13 + 1.62i)T + (5.08 + 18.3i)T^{2} \)
23 \( 1 + (-0.526 - 0.993i)T + (-12.9 + 19.0i)T^{2} \)
29 \( 1 + (5.88 - 3.98i)T + (10.7 - 26.9i)T^{2} \)
31 \( 1 + (2.13 + 2.80i)T + (-8.29 + 29.8i)T^{2} \)
37 \( 1 + (1.21 + 5.50i)T + (-33.5 + 15.5i)T^{2} \)
41 \( 1 + (-7.37 - 3.90i)T + (23.0 + 33.9i)T^{2} \)
43 \( 1 + (-2.77 - 2.92i)T + (-2.32 + 42.9i)T^{2} \)
47 \( 1 + (4.74 - 2.19i)T + (30.4 - 35.8i)T^{2} \)
53 \( 1 + (-6.69 + 11.1i)T + (-24.8 - 46.8i)T^{2} \)
61 \( 1 + (-1.44 - 0.982i)T + (22.5 + 56.6i)T^{2} \)
67 \( 1 + (-1.28 + 5.84i)T + (-60.8 - 28.1i)T^{2} \)
71 \( 1 + (5.62 + 12.1i)T + (-45.9 + 54.1i)T^{2} \)
73 \( 1 + (4.28 - 12.7i)T + (-58.1 - 44.1i)T^{2} \)
79 \( 1 + (-0.598 + 11.0i)T + (-78.5 - 8.54i)T^{2} \)
83 \( 1 + (-0.0783 + 0.477i)T + (-78.6 - 26.5i)T^{2} \)
89 \( 1 + (-8.74 - 12.8i)T + (-32.9 + 82.6i)T^{2} \)
97 \( 1 + (-1.72 - 5.11i)T + (-77.2 + 58.7i)T^{2} \)
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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

−12.94748112760570212468281476057, −12.05323000482760850897887864361, −11.15092932679986346715578303681, −9.374393743376216204054294315443, −8.918848439158901675324750240849, −7.78099125327945756144719026355, −6.84590928599788807397333667103, −5.92220912653203879941568668671, −5.03474582280226692144875049725, −1.81930332405076374372198741908, 0.789992979003020368788746515061, 3.09683148624293019467975315858, 4.15371061201763674353387573611, 5.78801225308030708124633709354, 7.50360093857693865788128506310, 8.654714024643198258109368720451, 9.905761803912946132863833374228, 10.47753311884323049934353512279, 10.90516543461116972036021672333, 11.96190203866308401296193511412

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