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Properties

Label 2-177-59.4-c1-0-1
Degree $2$
Conductor $177$
Sign $0.321 - 0.946i$
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  + (−2.53 − 0.276i)2-s + (−0.647 + 0.762i)3-s + (4.41 + 0.971i)4-s + (−0.0920 − 0.0699i)5-s + (1.85 − 1.75i)6-s + (0.00524 − 0.0131i)7-s + (−6.09 − 2.05i)8-s + (−0.161 − 0.986i)9-s + (0.214 + 0.203i)10-s + (0.676 − 0.312i)11-s + (−3.59 + 2.73i)12-s + (−0.486 + 2.96i)13-s + (−0.0169 + 0.0319i)14-s + (0.112 − 0.0248i)15-s + (6.69 + 3.09i)16-s + (2.12 + 5.32i)17-s + ⋯
L(s)  = 1  + (−1.79 − 0.195i)2-s + (−0.373 + 0.440i)3-s + (2.20 + 0.485i)4-s + (−0.0411 − 0.0313i)5-s + (0.756 − 0.716i)6-s + (0.00198 − 0.00497i)7-s + (−2.15 − 0.725i)8-s + (−0.0539 − 0.328i)9-s + (0.0677 + 0.0642i)10-s + (0.203 − 0.0943i)11-s + (−1.03 + 0.789i)12-s + (−0.134 + 0.822i)13-s + (−0.00452 + 0.00853i)14-s + (0.0291 − 0.00641i)15-s + (1.67 + 0.773i)16-s + (0.514 + 1.29i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.321 - 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.321 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.321 - 0.946i$
Analytic conductor: \(1.41335\)
Root analytic conductor: \(1.18884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :1/2),\ 0.321 - 0.946i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.340449 + 0.243827i\)
\(L(\frac12)\) \(\approx\) \(0.340449 + 0.243827i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.647 - 0.762i)T \)
59 \( 1 + (-4.98 - 5.84i)T \)
good2 \( 1 + (2.53 + 0.276i)T + (1.95 + 0.429i)T^{2} \)
5 \( 1 + (0.0920 + 0.0699i)T + (1.33 + 4.81i)T^{2} \)
7 \( 1 + (-0.00524 + 0.0131i)T + (-5.08 - 4.81i)T^{2} \)
11 \( 1 + (-0.676 + 0.312i)T + (7.12 - 8.38i)T^{2} \)
13 \( 1 + (0.486 - 2.96i)T + (-12.3 - 4.15i)T^{2} \)
17 \( 1 + (-2.12 - 5.32i)T + (-12.3 + 11.6i)T^{2} \)
19 \( 1 + (-3.20 - 4.72i)T + (-7.03 + 17.6i)T^{2} \)
23 \( 1 + (0.447 - 8.25i)T + (-22.8 - 2.48i)T^{2} \)
29 \( 1 + (-3.10 + 0.338i)T + (28.3 - 6.23i)T^{2} \)
31 \( 1 + (2.57 - 3.79i)T + (-11.4 - 28.7i)T^{2} \)
37 \( 1 + (4.76 - 1.60i)T + (29.4 - 22.3i)T^{2} \)
41 \( 1 + (0.347 + 6.40i)T + (-40.7 + 4.43i)T^{2} \)
43 \( 1 + (2.92 + 1.35i)T + (27.8 + 32.7i)T^{2} \)
47 \( 1 + (2.19 - 1.66i)T + (12.5 - 45.2i)T^{2} \)
53 \( 1 + (-7.37 + 6.99i)T + (2.86 - 52.9i)T^{2} \)
61 \( 1 + (-11.2 - 1.22i)T + (59.5 + 13.1i)T^{2} \)
67 \( 1 + (11.8 + 4.00i)T + (53.3 + 40.5i)T^{2} \)
71 \( 1 + (4.18 - 3.17i)T + (18.9 - 68.4i)T^{2} \)
73 \( 1 + (-7.00 + 13.2i)T + (-40.9 - 60.4i)T^{2} \)
79 \( 1 + (-3.27 - 3.85i)T + (-12.7 + 77.9i)T^{2} \)
83 \( 1 + (-10.3 + 6.20i)T + (38.8 - 73.3i)T^{2} \)
89 \( 1 + (15.7 - 1.71i)T + (86.9 - 19.1i)T^{2} \)
97 \( 1 + (-0.341 - 0.644i)T + (-54.4 + 80.2i)T^{2} \)
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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

−12.18074154578485670891169138135, −11.69424582383938441906492061685, −10.51525575722056024583019745682, −9.956149023098049834826905103585, −8.984547327343765757498556948909, −8.052450370372948214587154034937, −6.96499623328877657064403952341, −5.74185785042909429504675363836, −3.66291671408513940256450060248, −1.61668636094874377347565011579, 0.74954775525071641494299853459, 2.63383794426779568520892992426, 5.31866813723705222803243114297, 6.73601207616550874954429577093, 7.41639399906975479723870590767, 8.427491979929999808977648746596, 9.448190251617145725678774055007, 10.32155401604708466875447647860, 11.29455327378772560590123128435, 12.03553973668276724437544512101

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