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Properties

Label 2-74-37.11-c5-0-0
Degree $2$
Conductor $74$
Sign $-0.983 + 0.182i$
Analytic cond. $11.8684$
Root an. cond. $3.44505$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.46 − 2i)2-s + (−13.1 + 22.8i)3-s + (7.99 − 13.8i)4-s + (6.19 + 3.57i)5-s + 105. i·6-s + (−65.8 + 113. i)7-s − 63.9i·8-s + (−225. − 390. i)9-s + 28.6·10-s + 274.·11-s + (210. + 364. i)12-s + (−1.00e3 − 580. i)13-s + 526. i·14-s + (−163. + 94.1i)15-s + (−128 − 221. i)16-s + (−917. + 529. i)17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (−0.844 + 1.46i)3-s + (0.249 − 0.433i)4-s + (0.110 + 0.0639i)5-s + 1.19i·6-s + (−0.507 + 0.879i)7-s − 0.353i·8-s + (−0.927 − 1.60i)9-s + 0.0904·10-s + 0.684·11-s + (0.422 + 0.731i)12-s + (−1.64 − 0.952i)13-s + 0.717i·14-s + (−0.187 + 0.108i)15-s + (−0.125 − 0.216i)16-s + (−0.769 + 0.444i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 + 0.182i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.983 + 0.182i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $-0.983 + 0.182i$
Analytic conductor: \(11.8684\)
Root analytic conductor: \(3.44505\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{74} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 74,\ (\ :5/2),\ -0.983 + 0.182i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.0438262 - 0.475607i\)
\(L(\frac12)\) \(\approx\) \(0.0438262 - 0.475607i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-3.46 + 2i)T \)
37 \( 1 + (-6.70e3 - 4.93e3i)T \)
good3 \( 1 + (13.1 - 22.8i)T + (-121.5 - 210. i)T^{2} \)
5 \( 1 + (-6.19 - 3.57i)T + (1.56e3 + 2.70e3i)T^{2} \)
7 \( 1 + (65.8 - 113. i)T + (-8.40e3 - 1.45e4i)T^{2} \)
11 \( 1 - 274.T + 1.61e5T^{2} \)
13 \( 1 + (1.00e3 + 580. i)T + (1.85e5 + 3.21e5i)T^{2} \)
17 \( 1 + (917. - 529. i)T + (7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (785. + 453. i)T + (1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + 1.31e3iT - 6.43e6T^{2} \)
29 \( 1 - 8.64e3iT - 2.05e7T^{2} \)
31 \( 1 - 3.82e3iT - 2.86e7T^{2} \)
41 \( 1 + (-1.61e3 + 2.80e3i)T + (-5.79e7 - 1.00e8i)T^{2} \)
43 \( 1 + 196. iT - 1.47e8T^{2} \)
47 \( 1 + 1.42e4T + 2.29e8T^{2} \)
53 \( 1 + (-1.31e4 - 2.27e4i)T + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + (-1.92e4 + 1.11e4i)T + (3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (3.96e4 + 2.29e4i)T + (4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (2.46e4 - 4.26e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + (1.35e4 - 2.34e4i)T + (-9.02e8 - 1.56e9i)T^{2} \)
73 \( 1 + 3.45e4T + 2.07e9T^{2} \)
79 \( 1 + (-4.73e4 - 2.73e4i)T + (1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + (7.83e3 + 1.35e4i)T + (-1.96e9 + 3.41e9i)T^{2} \)
89 \( 1 + (7.36e4 - 4.25e4i)T + (2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 - 9.37e3iT - 8.58e9T^{2} \)
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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

−14.55927124798565555252428011579, −12.68161794400254520244070257014, −11.99494810267580037702984249574, −10.78193268664649320640833282935, −9.988626517039572818606522192377, −8.973195097703874465426549363568, −6.50838830519999766553704151514, −5.38107286918676080090645737383, −4.40053507374549297206253810479, −2.84194941927126723327717215010, 0.17622564336508065033469902577, 2.06151825398816352420026930194, 4.39071888334415692981642087452, 6.00361554950142517096698868124, 6.93588008157979861760329397333, 7.61064752707319991014266697647, 9.620641705400452034823514864994, 11.39717739252660316394466487787, 12.01235034678019582912668183730, 13.16558263480399808530063307257

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