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Properties

Label 2-74-37.10-c3-0-8
Degree $2$
Conductor $74$
Sign $-0.967 - 0.253i$
Analytic cond. $4.36614$
Root an. cond. $2.08953$
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  + (1 − 1.73i)2-s + (−4.31 − 7.47i)3-s + (−1.99 − 3.46i)4-s + (−2.90 − 5.02i)5-s − 17.2·6-s + (8.58 + 14.8i)7-s − 7.99·8-s + (−23.7 + 41.1i)9-s − 11.6·10-s + 9.76·11-s + (−17.2 + 29.8i)12-s + (−32.6 − 56.6i)13-s + 34.3·14-s + (−25.0 + 43.3i)15-s + (−8 + 13.8i)16-s + (12.0 − 20.9i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.830 − 1.43i)3-s + (−0.249 − 0.433i)4-s + (−0.259 − 0.449i)5-s − 1.17·6-s + (0.463 + 0.802i)7-s − 0.353·8-s + (−0.879 + 1.52i)9-s − 0.367·10-s + 0.267·11-s + (−0.415 + 0.719i)12-s + (−0.697 − 1.20i)13-s + 0.655·14-s + (−0.431 + 0.746i)15-s + (−0.125 + 0.216i)16-s + (0.172 − 0.298i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.967 - 0.253i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.967 - 0.253i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $-0.967 - 0.253i$
Analytic conductor: \(4.36614\)
Root analytic conductor: \(2.08953\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{74} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 74,\ (\ :3/2),\ -0.967 - 0.253i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.127216 + 0.988801i\)
\(L(\frac12)\) \(\approx\) \(0.127216 + 0.988801i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 + 1.73i)T \)
37 \( 1 + (-207. - 88.2i)T \)
good3 \( 1 + (4.31 + 7.47i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 + (2.90 + 5.02i)T + (-62.5 + 108. i)T^{2} \)
7 \( 1 + (-8.58 - 14.8i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 - 9.76T + 1.33e3T^{2} \)
13 \( 1 + (32.6 + 56.6i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (-12.0 + 20.9i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (34.7 + 60.1i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + 68.4T + 1.21e4T^{2} \)
29 \( 1 - 229.T + 2.43e4T^{2} \)
31 \( 1 + 271.T + 2.97e4T^{2} \)
41 \( 1 + (121. + 209. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 - 189.T + 7.95e4T^{2} \)
47 \( 1 - 506.T + 1.03e5T^{2} \)
53 \( 1 + (-183. + 317. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (444. - 769. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (303. + 524. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (125. + 217. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (-356. - 616. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + 87.8T + 3.89e5T^{2} \)
79 \( 1 + (216. + 375. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-455. + 789. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (370. - 642. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 1.38e3T + 9.12e5T^{2} \)
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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

−13.01638410685847622204192469124, −12.30241690008893502785802703363, −11.77205805629351696610442089904, −10.55234535549368132008498209709, −8.729163311996384696136802759395, −7.49992002737226943982856171891, −6.00723085955307961756035099300, −4.95955031312800757720647029800, −2.36490933685885891655351669252, −0.64192959471201016466670038523, 3.87674722550169716969886587031, 4.66641388026588970015683012684, 6.11254001409318395256754834415, 7.44616139705729291299212240666, 9.157582102801153355390293449967, 10.34127272756036106734370943465, 11.22027754723731760362875314774, 12.26595283303538525172227739099, 14.13755748503327359281876499569, 14.69842079538866438552244467867

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