# Properties

 Label 2-74-37.12-c3-0-4 Degree $2$ Conductor $74$ Sign $0.985 + 0.172i$ Analytic cond. $4.36614$ Root an. cond. $2.08953$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (1.53 − 1.28i)2-s + (4.80 + 4.03i)3-s + (0.694 − 3.93i)4-s + (14.5 − 5.31i)5-s + 12.5·6-s + (−23.2 + 8.47i)7-s + (−4.00 − 6.92i)8-s + (2.14 + 12.1i)9-s + (15.5 − 26.9i)10-s + (23.4 + 40.6i)11-s + (19.2 − 16.1i)12-s + (6.16 − 34.9i)13-s + (−24.7 + 42.9i)14-s + (91.5 + 33.3i)15-s + (−15.0 − 5.47i)16-s + (−5.54 − 31.4i)17-s + ⋯
 L(s)  = 1 + (0.541 − 0.454i)2-s + (0.924 + 0.775i)3-s + (0.0868 − 0.492i)4-s + (1.30 − 0.475i)5-s + 0.853·6-s + (−1.25 + 0.457i)7-s + (−0.176 − 0.306i)8-s + (0.0793 + 0.450i)9-s + (0.491 − 0.850i)10-s + (0.643 + 1.11i)11-s + (0.462 − 0.387i)12-s + (0.131 − 0.746i)13-s + (−0.473 + 0.819i)14-s + (1.57 + 0.573i)15-s + (−0.234 − 0.0855i)16-s + (−0.0791 − 0.448i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.172i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.985 + 0.172i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$74$$    =    $$2 \cdot 37$$ Sign: $0.985 + 0.172i$ Analytic conductor: $$4.36614$$ Root analytic conductor: $$2.08953$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{74} (49, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 74,\ (\ :3/2),\ 0.985 + 0.172i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$2.65480 - 0.230166i$$ $$L(\frac12)$$ $$\approx$$ $$2.65480 - 0.230166i$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (-1.53 + 1.28i)T$$
37 $$1 + (62.5 - 216. i)T$$
good3 $$1 + (-4.80 - 4.03i)T + (4.68 + 26.5i)T^{2}$$
5 $$1 + (-14.5 + 5.31i)T + (95.7 - 80.3i)T^{2}$$
7 $$1 + (23.2 - 8.47i)T + (262. - 220. i)T^{2}$$
11 $$1 + (-23.4 - 40.6i)T + (-665.5 + 1.15e3i)T^{2}$$
13 $$1 + (-6.16 + 34.9i)T + (-2.06e3 - 751. i)T^{2}$$
17 $$1 + (5.54 + 31.4i)T + (-4.61e3 + 1.68e3i)T^{2}$$
19 $$1 + (82.3 + 69.0i)T + (1.19e3 + 6.75e3i)T^{2}$$
23 $$1 + (90.0 - 156. i)T + (-6.08e3 - 1.05e4i)T^{2}$$
29 $$1 + (-50.8 - 88.0i)T + (-1.21e4 + 2.11e4i)T^{2}$$
31 $$1 + 138.T + 2.97e4T^{2}$$
41 $$1 + (-61.4 + 348. i)T + (-6.47e4 - 2.35e4i)T^{2}$$
43 $$1 + 342.T + 7.95e4T^{2}$$
47 $$1 + (-177. + 307. i)T + (-5.19e4 - 8.99e4i)T^{2}$$
53 $$1 + (-128. - 46.7i)T + (1.14e5 + 9.56e4i)T^{2}$$
59 $$1 + (20.6 + 7.50i)T + (1.57e5 + 1.32e5i)T^{2}$$
61 $$1 + (64.8 - 367. i)T + (-2.13e5 - 7.76e4i)T^{2}$$
67 $$1 + (-411. + 149. i)T + (2.30e5 - 1.93e5i)T^{2}$$
71 $$1 + (-575. - 483. i)T + (6.21e4 + 3.52e5i)T^{2}$$
73 $$1 - 587.T + 3.89e5T^{2}$$
79 $$1 + (-1.30e3 + 474. i)T + (3.77e5 - 3.16e5i)T^{2}$$
83 $$1 + (-122. - 693. i)T + (-5.37e5 + 1.95e5i)T^{2}$$
89 $$1 + (-52.5 - 19.1i)T + (5.40e5 + 4.53e5i)T^{2}$$
97 $$1 + (-367. + 636. i)T + (-4.56e5 - 7.90e5i)T^{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

−13.88289768052575127783067855572, −13.12236143740934862734379439478, −12.19545833501946284642746266078, −10.24925280545132311260563742347, −9.572325099573739104840928300599, −8.961376005007379034382139913312, −6.58953051262908800176339173568, −5.25204132241355992922291654875, −3.63877302453910346089480157960, −2.24335342343008486386971656391, 2.20599631443775439557033835428, 3.65452984224317874687363302477, 6.25642305622465118617359648393, 6.53484818696075412917746224694, 8.229088240209047745434322988666, 9.373483614837666893906434992092, 10.65664135009461238459130467313, 12.53919340396078427267314123358, 13.34708620688561228198604064075, 14.05297437608270219504760085514