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Properties

Label 2-74-37.6-c6-0-6
Degree $2$
Conductor $74$
Sign $0.216 - 0.976i$
Analytic cond. $17.0240$
Root an. cond. $4.12601$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4 − 4i)2-s + 44.2i·3-s − 32i·4-s + (111. + 111. i)5-s + (176. + 176. i)6-s + 600.·7-s + (−128 − 128i)8-s − 1.22e3·9-s + 892.·10-s + 246. i·11-s + 1.41e3·12-s + (−1.56e3 − 1.56e3i)13-s + (2.40e3 − 2.40e3i)14-s + (−4.93e3 + 4.93e3i)15-s − 1.02e3·16-s + (4.83e3 + 4.83e3i)17-s + ⋯
L(s)  = 1  + (0.5 − 0.5i)2-s + 1.63i·3-s − 0.5i·4-s + (0.892 + 0.892i)5-s + (0.819 + 0.819i)6-s + 1.75·7-s + (−0.250 − 0.250i)8-s − 1.68·9-s + 0.892·10-s + 0.185i·11-s + 0.819·12-s + (−0.710 − 0.710i)13-s + (0.875 − 0.875i)14-s + (−1.46 + 1.46i)15-s − 0.250·16-s + (0.984 + 0.984i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.216 - 0.976i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.216 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $0.216 - 0.976i$
Analytic conductor: \(17.0240\)
Root analytic conductor: \(4.12601\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{74} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 74,\ (\ :3),\ 0.216 - 0.976i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(2.45090 + 1.96732i\)
\(L(\frac12)\) \(\approx\) \(2.45090 + 1.96732i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-4 + 4i)T \)
37 \( 1 + (-6.84e3 + 5.01e4i)T \)
good3 \( 1 - 44.2iT - 729T^{2} \)
5 \( 1 + (-111. - 111. i)T + 1.56e4iT^{2} \)
7 \( 1 - 600.T + 1.17e5T^{2} \)
11 \( 1 - 246. iT - 1.77e6T^{2} \)
13 \( 1 + (1.56e3 + 1.56e3i)T + 4.82e6iT^{2} \)
17 \( 1 + (-4.83e3 - 4.83e3i)T + 2.41e7iT^{2} \)
19 \( 1 + (-4.17e3 - 4.17e3i)T + 4.70e7iT^{2} \)
23 \( 1 + (1.15e4 + 1.15e4i)T + 1.48e8iT^{2} \)
29 \( 1 + (2.79e4 - 2.79e4i)T - 5.94e8iT^{2} \)
31 \( 1 + (-7.33e3 + 7.33e3i)T - 8.87e8iT^{2} \)
41 \( 1 - 3.03e4iT - 4.75e9T^{2} \)
43 \( 1 + (3.25e4 + 3.25e4i)T + 6.32e9iT^{2} \)
47 \( 1 - 4.03e4T + 1.07e10T^{2} \)
53 \( 1 + 4.22e4T + 2.21e10T^{2} \)
59 \( 1 + (2.44e5 + 2.44e5i)T + 4.21e10iT^{2} \)
61 \( 1 + (-1.78e5 + 1.78e5i)T - 5.15e10iT^{2} \)
67 \( 1 + 2.37e5iT - 9.04e10T^{2} \)
71 \( 1 + 8.75e4T + 1.28e11T^{2} \)
73 \( 1 - 4.27e3iT - 1.51e11T^{2} \)
79 \( 1 + (-6.92e5 - 6.92e5i)T + 2.43e11iT^{2} \)
83 \( 1 - 1.05e6T + 3.26e11T^{2} \)
89 \( 1 + (-7.67e5 + 7.67e5i)T - 4.96e11iT^{2} \)
97 \( 1 + (3.75e5 + 3.75e5i)T + 8.32e11iT^{2} \)
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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

−14.23563705247843507265638503160, −12.27439913715338026943711107510, −10.93157794319461842657283756521, −10.46769908593066788361398621510, −9.606659976349966936897156810440, −7.984385813103392121915302390609, −5.74147343691040034465745111147, −4.91418063845559108053024705177, −3.57120116478672508507177814851, −2.02127728128792558252500134353, 1.14339753065479092802852044211, 2.15145677302847185075150745519, 4.91936666240411076312279904492, 5.78383618959173115653349784381, 7.35360092369894035094653672642, 8.004159197902486727059035357979, 9.320385986433173897951409996228, 11.67848515752641579867493425005, 12.02887702557157961558220259824, 13.60572126212573014867161282767

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