# Properties

 Label 2-74-37.31-c4-0-10 Degree $2$ Conductor $74$ Sign $-0.606 + 0.795i$ Analytic cond. $7.64937$ Root an. cond. $2.76575$ Motivic weight $4$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + (−2 − 2i)2-s − 10.5i·3-s + 8i·4-s + (30.2 − 30.2i)5-s + (−21.0 + 21.0i)6-s + 69.0·7-s + (16 − 16i)8-s − 29.6·9-s − 120.·10-s + 10.9i·11-s + 84.1·12-s + (80.0 − 80.0i)13-s + (−138. − 138. i)14-s + (−318. − 318. i)15-s − 64·16-s + (−345. + 345. i)17-s + ⋯
 L(s)  = 1 + (−0.5 − 0.5i)2-s − 1.16i·3-s + 0.5i·4-s + (1.20 − 1.20i)5-s + (−0.584 + 0.584i)6-s + 1.40·7-s + (0.250 − 0.250i)8-s − 0.365·9-s − 1.20·10-s + 0.0908i·11-s + 0.584·12-s + (0.473 − 0.473i)13-s + (−0.704 − 0.704i)14-s + (−1.41 − 1.41i)15-s − 0.250·16-s + (−1.19 + 1.19i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.606 + 0.795i)\, \overline{\Lambda}(5-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.606 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$74$$    =    $$2 \cdot 37$$ Sign: $-0.606 + 0.795i$ Analytic conductor: $$7.64937$$ Root analytic conductor: $$2.76575$$ Motivic weight: $$4$$ Rational: no Arithmetic: yes Character: $\chi_{74} (31, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 74,\ (\ :2),\ -0.606 + 0.795i)$$

## Particular Values

 $$L(\frac{5}{2})$$ $$\approx$$ $$0.774812 - 1.56528i$$ $$L(\frac12)$$ $$\approx$$ $$0.774812 - 1.56528i$$ $$L(3)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (2 + 2i)T$$
37 $$1 + (917. - 1.01e3i)T$$
good3 $$1 + 10.5iT - 81T^{2}$$
5 $$1 + (-30.2 + 30.2i)T - 625iT^{2}$$
7 $$1 - 69.0T + 2.40e3T^{2}$$
11 $$1 - 10.9iT - 1.46e4T^{2}$$
13 $$1 + (-80.0 + 80.0i)T - 2.85e4iT^{2}$$
17 $$1 + (345. - 345. i)T - 8.35e4iT^{2}$$
19 $$1 + (-121. + 121. i)T - 1.30e5iT^{2}$$
23 $$1 + (590. - 590. i)T - 2.79e5iT^{2}$$
29 $$1 + (-49.9 - 49.9i)T + 7.07e5iT^{2}$$
31 $$1 + (-947. - 947. i)T + 9.23e5iT^{2}$$
41 $$1 - 3.12e3iT - 2.82e6T^{2}$$
43 $$1 + (-522. + 522. i)T - 3.41e6iT^{2}$$
47 $$1 - 315.T + 4.87e6T^{2}$$
53 $$1 + 3.94e3T + 7.89e6T^{2}$$
59 $$1 + (-3.55e3 + 3.55e3i)T - 1.21e7iT^{2}$$
61 $$1 + (519. + 519. i)T + 1.38e7iT^{2}$$
67 $$1 + 3.73e3iT - 2.01e7T^{2}$$
71 $$1 + 1.19e3T + 2.54e7T^{2}$$
73 $$1 + 1.21e3iT - 2.83e7T^{2}$$
79 $$1 + (1.83e3 - 1.83e3i)T - 3.89e7iT^{2}$$
83 $$1 + 1.53e3T + 4.74e7T^{2}$$
89 $$1 + (5.11e3 + 5.11e3i)T + 6.27e7iT^{2}$$
97 $$1 + (4.01e3 - 4.01e3i)T - 8.85e7iT^{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

−13.27483033667788077492938058178, −12.46749536524070560896133116661, −11.37077397424118757518563024589, −10.00554483299274852635592476427, −8.617316953628916391394798707416, −7.990308326956519671556349567688, −6.26050823515227891069350370986, −4.78432131872727004039885877458, −1.86728897107088372558998871780, −1.28547910699209713373851189589, 2.17613300678973335575849843622, 4.44927885833823026004442366468, 5.79137329609808236942747385914, 7.11188818069892158041418494378, 8.722503602105646054259797566138, 9.795213792714066640640965797795, 10.63500572597066570318911233270, 11.38366178946991636486286455645, 13.87976175284747651376862208975, 14.27133553058726278660950425593