Properties

Label 2-177-177.83-c1-0-16
Degree $2$
Conductor $177$
Sign $-0.840 + 0.541i$
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  + (−0.487 − 1.75i)2-s + (1.13 − 1.30i)3-s + (−1.13 + 0.683i)4-s + (−0.529 + 0.0287i)5-s + (−2.85 − 1.35i)6-s + (0.0261 + 0.159i)7-s + (−0.892 − 0.845i)8-s + (−0.418 − 2.97i)9-s + (0.308 + 0.916i)10-s + (−0.214 − 0.405i)11-s + (−0.397 + 2.26i)12-s + (1.99 + 0.795i)13-s + (0.267 − 0.123i)14-s + (−0.564 + 0.724i)15-s + (−2.29 + 4.32i)16-s + (2.58 + 0.423i)17-s + ⋯
L(s)  = 1  + (−0.344 − 1.24i)2-s + (0.655 − 0.754i)3-s + (−0.568 + 0.341i)4-s + (−0.236 + 0.0128i)5-s + (−1.16 − 0.554i)6-s + (0.00988 + 0.0602i)7-s + (−0.315 − 0.298i)8-s + (−0.139 − 0.990i)9-s + (0.0976 + 0.289i)10-s + (−0.0647 − 0.122i)11-s + (−0.114 + 0.652i)12-s + (0.553 + 0.220i)13-s + (0.0714 − 0.0330i)14-s + (−0.145 + 0.187i)15-s + (−0.573 + 1.08i)16-s + (0.625 + 0.102i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.840 + 0.541i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.840 + 0.541i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-0.840 + 0.541i$
Analytic conductor: \(1.41335\)
Root analytic conductor: \(1.18884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :1/2),\ -0.840 + 0.541i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.323393 - 1.09874i\)
\(L(\frac12)\) \(\approx\) \(0.323393 - 1.09874i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.13 + 1.30i)T \)
59 \( 1 + (-1.40 - 7.55i)T \)
good2 \( 1 + (0.487 + 1.75i)T + (-1.71 + 1.03i)T^{2} \)
5 \( 1 + (0.529 - 0.0287i)T + (4.97 - 0.540i)T^{2} \)
7 \( 1 + (-0.0261 - 0.159i)T + (-6.63 + 2.23i)T^{2} \)
11 \( 1 + (0.214 + 0.405i)T + (-6.17 + 9.10i)T^{2} \)
13 \( 1 + (-1.99 - 0.795i)T + (9.43 + 8.94i)T^{2} \)
17 \( 1 + (-2.58 - 0.423i)T + (16.1 + 5.42i)T^{2} \)
19 \( 1 + (0.426 - 0.501i)T + (-3.07 - 18.7i)T^{2} \)
23 \( 1 + (-4.86 - 3.70i)T + (6.15 + 22.1i)T^{2} \)
29 \( 1 + (-4.99 - 1.38i)T + (24.8 + 14.9i)T^{2} \)
31 \( 1 + (-3.15 + 2.68i)T + (5.01 - 30.5i)T^{2} \)
37 \( 1 + (0.125 + 0.132i)T + (-2.00 + 36.9i)T^{2} \)
41 \( 1 + (4.26 + 5.61i)T + (-10.9 + 39.5i)T^{2} \)
43 \( 1 + (-5.67 - 3.00i)T + (24.1 + 35.5i)T^{2} \)
47 \( 1 + (0.327 - 6.03i)T + (-46.7 - 5.08i)T^{2} \)
53 \( 1 + (-2.13 + 6.34i)T + (-42.1 - 32.0i)T^{2} \)
61 \( 1 + (7.50 - 2.08i)T + (52.2 - 31.4i)T^{2} \)
67 \( 1 + (2.85 - 3.01i)T + (-3.62 - 66.9i)T^{2} \)
71 \( 1 + (7.34 + 0.398i)T + (70.5 + 7.67i)T^{2} \)
73 \( 1 + (-0.944 - 2.04i)T + (-47.2 + 55.6i)T^{2} \)
79 \( 1 + (3.58 + 5.29i)T + (-29.2 + 73.3i)T^{2} \)
83 \( 1 + (14.1 - 3.10i)T + (75.3 - 34.8i)T^{2} \)
89 \( 1 + (-1.67 + 6.03i)T + (-76.2 - 45.8i)T^{2} \)
97 \( 1 + (7.32 - 15.8i)T + (-62.7 - 73.9i)T^{2} \)
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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

−12.11357019837138083570644718513, −11.49652539278517019935635191149, −10.35743607346450209103712721552, −9.315890388233451437507480728942, −8.473239874283275556225342364337, −7.29419248415899210419772896460, −6.01918054149836459595905141598, −3.85068163256937894297989860232, −2.75948228787790467568074942480, −1.28421313736744790508747269276, 2.95643503918543532279797749055, 4.53621010672237704197862592589, 5.75317544150948929630333931889, 7.06045953300059112195902052851, 8.101361409469696507386347304165, 8.726396401870525506570852860875, 9.812049846994661677873888627032, 10.87018050428695447052056855527, 12.10633859002865221988320867928, 13.55949880724784149232617414946

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