Properties

Label 2-177-59.51-c1-0-4
Degree $2$
Conductor $177$
Sign $0.671 + 0.740i$
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  + (0.302 − 1.84i)2-s + (−0.267 + 0.963i)3-s + (−1.41 − 0.478i)4-s + (1.62 + 2.39i)5-s + (1.69 + 0.785i)6-s + (1.15 − 0.255i)7-s + (0.440 − 0.830i)8-s + (−0.856 − 0.515i)9-s + (4.90 − 2.26i)10-s + (3.14 − 2.39i)11-s + (0.840 − 1.23i)12-s + (−3.86 + 2.32i)13-s + (−0.120 − 2.21i)14-s + (−2.73 + 0.922i)15-s + (−3.78 − 2.87i)16-s + (0.853 + 0.187i)17-s + ⋯
L(s)  = 1  + (0.213 − 1.30i)2-s + (−0.154 + 0.556i)3-s + (−0.709 − 0.239i)4-s + (0.724 + 1.06i)5-s + (0.692 + 0.320i)6-s + (0.438 − 0.0964i)7-s + (0.155 − 0.293i)8-s + (−0.285 − 0.171i)9-s + (1.55 − 0.717i)10-s + (0.948 − 0.721i)11-s + (0.242 − 0.357i)12-s + (−1.07 + 0.644i)13-s + (−0.0321 − 0.592i)14-s + (−0.706 + 0.238i)15-s + (−0.945 − 0.719i)16-s + (0.207 + 0.0455i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.671 + 0.740i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.671 + 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.671 + 0.740i$
Analytic conductor: \(1.41335\)
Root analytic conductor: \(1.18884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :1/2),\ 0.671 + 0.740i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.30100 - 0.576309i\)
\(L(\frac12)\) \(\approx\) \(1.30100 - 0.576309i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.267 - 0.963i)T \)
59 \( 1 + (3.84 - 6.65i)T \)
good2 \( 1 + (-0.302 + 1.84i)T + (-1.89 - 0.638i)T^{2} \)
5 \( 1 + (-1.62 - 2.39i)T + (-1.85 + 4.64i)T^{2} \)
7 \( 1 + (-1.15 + 0.255i)T + (6.35 - 2.93i)T^{2} \)
11 \( 1 + (-3.14 + 2.39i)T + (2.94 - 10.5i)T^{2} \)
13 \( 1 + (3.86 - 2.32i)T + (6.08 - 11.4i)T^{2} \)
17 \( 1 + (-0.853 - 0.187i)T + (15.4 + 7.13i)T^{2} \)
19 \( 1 + (3.75 - 0.408i)T + (18.5 - 4.08i)T^{2} \)
23 \( 1 + (-2.31 - 2.73i)T + (-3.72 + 22.6i)T^{2} \)
29 \( 1 + (0.671 + 4.09i)T + (-27.4 + 9.25i)T^{2} \)
31 \( 1 + (5.65 + 0.614i)T + (30.2 + 6.66i)T^{2} \)
37 \( 1 + (2.58 + 4.87i)T + (-20.7 + 30.6i)T^{2} \)
41 \( 1 + (4.30 - 5.06i)T + (-6.63 - 40.4i)T^{2} \)
43 \( 1 + (-7.31 - 5.55i)T + (11.5 + 41.4i)T^{2} \)
47 \( 1 + (-5.31 + 7.83i)T + (-17.3 - 43.6i)T^{2} \)
53 \( 1 + (9.18 + 4.25i)T + (34.3 + 40.3i)T^{2} \)
61 \( 1 + (1.88 - 11.5i)T + (-57.8 - 19.4i)T^{2} \)
67 \( 1 + (0.875 - 1.65i)T + (-37.5 - 55.4i)T^{2} \)
71 \( 1 + (-2.55 + 3.76i)T + (-26.2 - 65.9i)T^{2} \)
73 \( 1 + (0.451 + 8.31i)T + (-72.5 + 7.89i)T^{2} \)
79 \( 1 + (3.71 + 13.3i)T + (-67.6 + 40.7i)T^{2} \)
83 \( 1 + (-11.1 - 10.5i)T + (4.49 + 82.8i)T^{2} \)
89 \( 1 + (0.307 + 1.87i)T + (-84.3 + 28.4i)T^{2} \)
97 \( 1 + (-0.436 + 8.04i)T + (-96.4 - 10.4i)T^{2} \)
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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

−12.23901653960298159516816423934, −11.34657467175182376544302560460, −10.79051521699791827671419102202, −9.881526100752583915822194189943, −9.110309177724812290454300028942, −7.22684370243229054463140763325, −6.07281896777399558816376800853, −4.48154594994766287306576874694, −3.29810081308231731247320372670, −2.00927962582800908760034307963, 1.85323001669890736194250470968, 4.70992013647189329525229190203, 5.38991135526924803802629407265, 6.54759986905108544138475715031, 7.46710008545126669156461849395, 8.541824284176575779239807678071, 9.398117445285057895878181530093, 10.88226823691577074881716942585, 12.37019075529938483711994977873, 12.78413114556473454763545741619

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