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Properties

Label 2-174-3.2-c4-0-5
Degree $2$
Conductor $174$
Sign $-0.160 + 0.987i$
Analytic cond. $17.9863$
Root an. cond. $4.24103$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.82i·2-s + (−1.44 + 8.88i)3-s − 8.00·4-s + 16.8i·5-s + (−25.1 − 4.08i)6-s + 24.6·7-s − 22.6i·8-s + (−76.8 − 25.6i)9-s − 47.5·10-s + 133. i·11-s + (11.5 − 71.0i)12-s − 268.·13-s + 69.6i·14-s + (−149. − 24.2i)15-s + 64.0·16-s + 19.3i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.160 + 0.987i)3-s − 0.500·4-s + 0.672i·5-s + (−0.697 − 0.113i)6-s + 0.502·7-s − 0.353i·8-s + (−0.948 − 0.316i)9-s − 0.475·10-s + 1.10i·11-s + (0.0801 − 0.493i)12-s − 1.59·13-s + 0.355i·14-s + (−0.663 − 0.107i)15-s + 0.250·16-s + 0.0669i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 174 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.160 + 0.987i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 174 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.160 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(174\)    =    \(2 \cdot 3 \cdot 29\)
Sign: $-0.160 + 0.987i$
Analytic conductor: \(17.9863\)
Root analytic conductor: \(4.24103\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{174} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 174,\ (\ :2),\ -0.160 + 0.987i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.6052253291\)
\(L(\frac12)\) \(\approx\) \(0.6052253291\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2.82iT \)
3 \( 1 + (1.44 - 8.88i)T \)
29 \( 1 + 156. iT \)
good5 \( 1 - 16.8iT - 625T^{2} \)
7 \( 1 - 24.6T + 2.40e3T^{2} \)
11 \( 1 - 133. iT - 1.46e4T^{2} \)
13 \( 1 + 268.T + 2.85e4T^{2} \)
17 \( 1 - 19.3iT - 8.35e4T^{2} \)
19 \( 1 + 180.T + 1.30e5T^{2} \)
23 \( 1 + 534. iT - 2.79e5T^{2} \)
31 \( 1 - 557.T + 9.23e5T^{2} \)
37 \( 1 + 857.T + 1.87e6T^{2} \)
41 \( 1 - 1.04e3iT - 2.82e6T^{2} \)
43 \( 1 - 1.23e3T + 3.41e6T^{2} \)
47 \( 1 + 1.08e3iT - 4.87e6T^{2} \)
53 \( 1 + 485. iT - 7.89e6T^{2} \)
59 \( 1 + 480. iT - 1.21e7T^{2} \)
61 \( 1 - 1.22e3T + 1.38e7T^{2} \)
67 \( 1 + 5.97e3T + 2.01e7T^{2} \)
71 \( 1 - 4.52e3iT - 2.54e7T^{2} \)
73 \( 1 + 9.48e3T + 2.83e7T^{2} \)
79 \( 1 + 6.55e3T + 3.89e7T^{2} \)
83 \( 1 - 8.58e3iT - 4.74e7T^{2} \)
89 \( 1 + 4.97e3iT - 6.27e7T^{2} \)
97 \( 1 - 1.21e4T + 8.85e7T^{2} \)
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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

−12.70001842112643382538178224226, −11.71711067042637259829252852228, −10.44927481205076675906475468628, −9.887922553748448444229404880612, −8.726484109539774986521496029405, −7.50482317581838984008869198331, −6.50408513607201552436340500376, −5.05494309106708852455162756509, −4.35664573991886906466488130739, −2.60943408127550592644379813260, 0.22874922268086099495557402187, 1.51888182262746866634827442656, 2.88715389630054183987577364973, 4.73657240099845603215051444284, 5.74545981373864040676451097506, 7.29297267821730082393006401311, 8.305286102565350129407002028998, 9.172304199638315786871296522452, 10.57144291035752808880383916824, 11.58495131710777250756563719491

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