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Properties

Label 2-74-37.36-c3-0-3
Degree $2$
Conductor $74$
Sign $0.939 + 0.343i$
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 − 8.15·3-s − 4·4-s − 6.18i·5-s − 16.3i·6-s + 23.1·7-s − 8i·8-s + 39.5·9-s + 12.3·10-s + 13.1·11-s + 32.6·12-s − 30.8i·13-s + 46.3i·14-s + 50.4i·15-s + 16·16-s − 87.4i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s − 1.56·3-s − 0.5·4-s − 0.553i·5-s − 1.10i·6-s + 1.25·7-s − 0.353i·8-s + 1.46·9-s + 0.391·10-s + 0.361·11-s + 0.784·12-s − 0.657i·13-s + 0.884i·14-s + 0.868i·15-s + 0.250·16-s − 1.24i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.343i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.939 + 0.343i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $0.939 + 0.343i$
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: \((2,\ 74,\ (\ :3/2),\ 0.939 + 0.343i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.857122 - 0.151749i\)
\(L(\frac12)\) \(\approx\) \(0.857122 - 0.151749i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-77.2 + 211. i)T \)
good3 \( 1 + 8.15T + 27T^{2} \)
5 \( 1 + 6.18iT - 125T^{2} \)
7 \( 1 - 23.1T + 343T^{2} \)
11 \( 1 - 13.1T + 1.33e3T^{2} \)
13 \( 1 + 30.8iT - 2.19e3T^{2} \)
17 \( 1 + 87.4iT - 4.91e3T^{2} \)
19 \( 1 + 93.7iT - 6.85e3T^{2} \)
23 \( 1 - 73.1iT - 1.21e4T^{2} \)
29 \( 1 + 199. iT - 2.43e4T^{2} \)
31 \( 1 - 125. iT - 2.97e4T^{2} \)
41 \( 1 + 339.T + 6.89e4T^{2} \)
43 \( 1 - 501. iT - 7.95e4T^{2} \)
47 \( 1 - 282.T + 1.03e5T^{2} \)
53 \( 1 + 205.T + 1.48e5T^{2} \)
59 \( 1 + 680. iT - 2.05e5T^{2} \)
61 \( 1 + 352. iT - 2.26e5T^{2} \)
67 \( 1 - 739.T + 3.00e5T^{2} \)
71 \( 1 + 188.T + 3.57e5T^{2} \)
73 \( 1 - 1.21e3T + 3.89e5T^{2} \)
79 \( 1 + 750. iT - 4.93e5T^{2} \)
83 \( 1 + 982.T + 5.71e5T^{2} \)
89 \( 1 - 1.14e3iT - 7.04e5T^{2} \)
97 \( 1 - 273. iT - 9.12e5T^{2} \)
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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

−14.07072193409362905170672881271, −12.82229559796990474842816821467, −11.68935387560709535561843872295, −10.98779776503868100305422954154, −9.440300164216397805289130676939, −7.978258393627060696577174293603, −6.71065323522841828414438851097, −5.29078506098441040173064101678, −4.75048798126884956137107331155, −0.78150981503872105537388907560, 1.52035539451378723934098670060, 4.19489515334973722539682901661, 5.46660725575120766707499914644, 6.77272909067379132015194232665, 8.456578556324302697332629370991, 10.29754688155921491690180761448, 10.91152338556818413380251076699, 11.77312315197602172918101930830, 12.52108142832861190431183884446, 14.12611708048510288274238771217

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