Properties

Label 2-177-59.4-c3-0-27
Degree $2$
Conductor $177$
Sign $-0.898 + 0.438i$
Analytic cond. $10.4433$
Root an. cond. $3.23161$
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  + (2.00 + 0.217i)2-s + (1.94 − 2.28i)3-s + (−3.85 − 0.848i)4-s + (−3.87 − 2.94i)5-s + (4.38 − 4.15i)6-s + (5.75 − 14.4i)7-s + (−22.7 − 7.67i)8-s + (−1.45 − 8.88i)9-s + (−7.10 − 6.73i)10-s + (−55.5 + 25.6i)11-s + (−9.42 + 7.16i)12-s + (−6.50 + 39.6i)13-s + (14.6 − 27.6i)14-s + (−14.2 + 3.13i)15-s + (−15.2 − 7.06i)16-s + (−12.6 − 31.6i)17-s + ⋯
L(s)  = 1  + (0.707 + 0.0769i)2-s + (0.373 − 0.440i)3-s + (−0.481 − 0.106i)4-s + (−0.346 − 0.263i)5-s + (0.298 − 0.282i)6-s + (0.310 − 0.779i)7-s + (−1.00 − 0.339i)8-s + (−0.0539 − 0.328i)9-s + (−0.224 − 0.212i)10-s + (−1.52 + 0.704i)11-s + (−0.226 + 0.172i)12-s + (−0.138 + 0.846i)13-s + (0.279 − 0.527i)14-s + (−0.245 + 0.0539i)15-s + (−0.238 − 0.110i)16-s + (−0.179 − 0.451i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.898 + 0.438i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.898 + 0.438i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-0.898 + 0.438i$
Analytic conductor: \(10.4433\)
Root analytic conductor: \(3.23161\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :3/2),\ -0.898 + 0.438i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.239070 - 1.03463i\)
\(L(\frac12)\) \(\approx\) \(0.239070 - 1.03463i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.94 + 2.28i)T \)
59 \( 1 + (-37.1 + 451. i)T \)
good2 \( 1 + (-2.00 - 0.217i)T + (7.81 + 1.71i)T^{2} \)
5 \( 1 + (3.87 + 2.94i)T + (33.4 + 120. i)T^{2} \)
7 \( 1 + (-5.75 + 14.4i)T + (-249. - 235. i)T^{2} \)
11 \( 1 + (55.5 - 25.6i)T + (861. - 1.01e3i)T^{2} \)
13 \( 1 + (6.50 - 39.6i)T + (-2.08e3 - 701. i)T^{2} \)
17 \( 1 + (12.6 + 31.6i)T + (-3.56e3 + 3.37e3i)T^{2} \)
19 \( 1 + (32.7 + 48.3i)T + (-2.53e3 + 6.37e3i)T^{2} \)
23 \( 1 + (-7.86 + 145. i)T + (-1.20e4 - 1.31e3i)T^{2} \)
29 \( 1 + (-57.7 + 6.27i)T + (2.38e4 - 5.24e3i)T^{2} \)
31 \( 1 + (20.3 - 30.0i)T + (-1.10e4 - 2.76e4i)T^{2} \)
37 \( 1 + (-183. + 61.8i)T + (4.03e4 - 3.06e4i)T^{2} \)
41 \( 1 + (7.42 + 137. i)T + (-6.85e4 + 7.45e3i)T^{2} \)
43 \( 1 + (-43.8 - 20.2i)T + (5.14e4 + 6.05e4i)T^{2} \)
47 \( 1 + (154. - 117. i)T + (2.77e4 - 1.00e5i)T^{2} \)
53 \( 1 + (214. - 202. i)T + (8.06e3 - 1.48e5i)T^{2} \)
61 \( 1 + (-48.6 - 5.29i)T + (2.21e5 + 4.87e4i)T^{2} \)
67 \( 1 + (-277. - 93.6i)T + (2.39e5 + 1.82e5i)T^{2} \)
71 \( 1 + (111. - 84.9i)T + (9.57e4 - 3.44e5i)T^{2} \)
73 \( 1 + (-482. + 909. i)T + (-2.18e5 - 3.21e5i)T^{2} \)
79 \( 1 + (-95.1 - 111. i)T + (-7.97e4 + 4.86e5i)T^{2} \)
83 \( 1 + (410. - 247. i)T + (2.67e5 - 5.05e5i)T^{2} \)
89 \( 1 + (-312. + 33.9i)T + (6.88e5 - 1.51e5i)T^{2} \)
97 \( 1 + (359. + 677. i)T + (-5.12e5 + 7.55e5i)T^{2} \)
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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

−12.26211083168313885520639134596, −10.89114718741069545464569685045, −9.764599278967678360678987533866, −8.612321586349803241875877493631, −7.63294705358076846017687018137, −6.51863321547371280724256164542, −4.89479651689189994235603741972, −4.26133024612860127742465009351, −2.54831821668756117957489907674, −0.35412733820155459883004045117, 2.70037507900270806140154209678, 3.66596944147761938470927353016, 5.12200577117794816353576714534, 5.76606575357349645278975932401, 7.86853913940892679404229013353, 8.431968055198679657912500468192, 9.649838944729180270897460479108, 10.78058045874423467372598701995, 11.76194882345788502398287450945, 12.94195211324460909438787309804

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