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Properties

Label 2-74-37.36-c3-0-0
Degree $2$
Conductor $74$
Sign $-0.992 + 0.124i$
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  + 2i·2-s − 1.87·3-s − 4·4-s + 8.73i·5-s − 3.74i·6-s − 6.27·7-s − 8i·8-s − 23.5·9-s − 17.4·10-s − 45.9·11-s + 7.48·12-s + 7.07i·13-s − 12.5i·14-s − 16.3i·15-s + 16·16-s + 36.5i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.359·3-s − 0.5·4-s + 0.781i·5-s − 0.254i·6-s − 0.338·7-s − 0.353i·8-s − 0.870·9-s − 0.552·10-s − 1.25·11-s + 0.179·12-s + 0.150i·13-s − 0.239i·14-s − 0.281i·15-s + 0.250·16-s + 0.521i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 + 0.124i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.992 + 0.124i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $-0.992 + 0.124i$
Analytic conductor: \(4.36614\)
Root analytic conductor: \(2.08953\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{74} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 74,\ (\ :3/2),\ -0.992 + 0.124i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0341148 - 0.547453i\)
\(L(\frac12)\) \(\approx\) \(0.0341148 - 0.547453i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2iT \)
37 \( 1 + (-27.9 - 223. i)T \)
good3 \( 1 + 1.87T + 27T^{2} \)
5 \( 1 - 8.73iT - 125T^{2} \)
7 \( 1 + 6.27T + 343T^{2} \)
11 \( 1 + 45.9T + 1.33e3T^{2} \)
13 \( 1 - 7.07iT - 2.19e3T^{2} \)
17 \( 1 - 36.5iT - 4.91e3T^{2} \)
19 \( 1 + 20.9iT - 6.85e3T^{2} \)
23 \( 1 - 89.2iT - 1.21e4T^{2} \)
29 \( 1 - 223. iT - 2.43e4T^{2} \)
31 \( 1 + 18.0iT - 2.97e4T^{2} \)
41 \( 1 - 101.T + 6.89e4T^{2} \)
43 \( 1 + 255. iT - 7.95e4T^{2} \)
47 \( 1 - 288.T + 1.03e5T^{2} \)
53 \( 1 + 368.T + 1.48e5T^{2} \)
59 \( 1 + 211. iT - 2.05e5T^{2} \)
61 \( 1 - 178. iT - 2.26e5T^{2} \)
67 \( 1 + 773.T + 3.00e5T^{2} \)
71 \( 1 - 288.T + 3.57e5T^{2} \)
73 \( 1 + 442.T + 3.89e5T^{2} \)
79 \( 1 + 299. iT - 4.93e5T^{2} \)
83 \( 1 + 490.T + 5.71e5T^{2} \)
89 \( 1 - 1.44e3iT - 7.04e5T^{2} \)
97 \( 1 + 306. iT - 9.12e5T^{2} \)
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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

−14.73722198828532263313578090451, −13.73784179163372344456435203110, −12.62527401542017348057375198579, −11.17509939690156041498997231998, −10.26114284922037717162578904541, −8.772036873729357876042294757526, −7.49477314386140055239284830784, −6.30741301579360555753406099915, −5.16923778750482219107148364834, −3.08140333921697894308466505557, 0.35307995440977658816849724499, 2.74557956493813177053342022186, 4.71240654897176763884341170451, 5.87390003091702978284351591934, 7.924571441381182997584641674647, 9.039923512835522355221129350037, 10.28367526247995025261508414942, 11.32912011734373793134248771617, 12.43796672551720531753430674760, 13.17799894446404761758795756509

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