# Properties

 Label 2-78-39.20-c1-0-0 Degree $2$ Conductor $78$ Sign $-0.373 - 0.927i$ Analytic cond. $0.622833$ Root an. cond. $0.789197$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.258 + 0.965i)2-s + (−1.45 + 0.933i)3-s + (−0.866 + 0.499i)4-s + (−2.02 + 2.02i)5-s + (−1.27 − 1.16i)6-s + (3.46 + 0.929i)7-s + (−0.707 − 0.707i)8-s + (1.25 − 2.72i)9-s + (−2.47 − 1.42i)10-s + (4.05 − 1.08i)11-s + (0.796 − 1.53i)12-s + (−3.60 + 0.176i)13-s + 3.58i·14-s + (1.06 − 4.83i)15-s + (0.500 − 0.866i)16-s + (1.72 + 2.99i)17-s + ⋯
 L(s)  = 1 + (0.183 + 0.683i)2-s + (−0.842 + 0.539i)3-s + (−0.433 + 0.249i)4-s + (−0.903 + 0.903i)5-s + (−0.522 − 0.476i)6-s + (1.31 + 0.351i)7-s + (−0.249 − 0.249i)8-s + (0.418 − 0.908i)9-s + (−0.782 − 0.451i)10-s + (1.22 − 0.327i)11-s + (0.229 − 0.444i)12-s + (−0.998 + 0.0490i)13-s + 0.959i·14-s + (0.273 − 1.24i)15-s + (0.125 − 0.216i)16-s + (0.418 + 0.725i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.373 - 0.927i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.373 - 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$78$$    =    $$2 \cdot 3 \cdot 13$$ Sign: $-0.373 - 0.927i$ Analytic conductor: $$0.622833$$ Root analytic conductor: $$0.789197$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{78} (59, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 78,\ (\ :1/2),\ -0.373 - 0.927i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.433217 + 0.641344i$$ $$L(\frac12)$$ $$\approx$$ $$0.433217 + 0.641344i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (-0.258 - 0.965i)T$$
3 $$1 + (1.45 - 0.933i)T$$
13 $$1 + (3.60 - 0.176i)T$$
good5 $$1 + (2.02 - 2.02i)T - 5iT^{2}$$
7 $$1 + (-3.46 - 0.929i)T + (6.06 + 3.5i)T^{2}$$
11 $$1 + (-4.05 + 1.08i)T + (9.52 - 5.5i)T^{2}$$
17 $$1 + (-1.72 - 2.99i)T + (-8.5 + 14.7i)T^{2}$$
19 $$1 + (-0.581 + 2.16i)T + (-16.4 - 9.5i)T^{2}$$
23 $$1 + (-1.51 + 2.62i)T + (-11.5 - 19.9i)T^{2}$$
29 $$1 + (1.74 + 1.00i)T + (14.5 + 25.1i)T^{2}$$
31 $$1 + (-1.21 - 1.21i)T + 31iT^{2}$$
37 $$1 + (1.25 + 4.67i)T + (-32.0 + 18.5i)T^{2}$$
41 $$1 + (2.05 + 7.67i)T + (-35.5 + 20.5i)T^{2}$$
43 $$1 + (1.68 - 0.975i)T + (21.5 - 37.2i)T^{2}$$
47 $$1 + (-0.957 - 0.957i)T + 47iT^{2}$$
53 $$1 - 7.22iT - 53T^{2}$$
59 $$1 + (-2.66 + 9.93i)T + (-51.0 - 29.5i)T^{2}$$
61 $$1 + (0.137 + 0.237i)T + (-30.5 + 52.8i)T^{2}$$
67 $$1 + (-4.10 + 1.09i)T + (58.0 - 33.5i)T^{2}$$
71 $$1 + (10.6 + 2.85i)T + (61.4 + 35.5i)T^{2}$$
73 $$1 + (-10.0 + 10.0i)T - 73iT^{2}$$
79 $$1 - 1.58T + 79T^{2}$$
83 $$1 + (-2.58 + 2.58i)T - 83iT^{2}$$
89 $$1 + (9.50 - 2.54i)T + (77.0 - 44.5i)T^{2}$$
97 $$1 + (2.07 - 7.74i)T + (-84.0 - 48.5i)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

−14.81991138872175130034663368917, −14.36683371845398085121202802019, −12.26668620455312118614150866995, −11.56510017879757605075435837467, −10.65229074303146141476932284530, −9.051330154245990281004535961796, −7.65262014659727648802297335759, −6.54990046093244308338979486406, −5.09230385833849599776583081698, −3.86856995791394296230245012185, 1.31994732191858571724124188623, 4.33861600746475468326518792472, 5.17896898050742297987606934607, 7.23319642229864062345135020098, 8.293483157278294185601681381058, 9.868658593760215567117000642475, 11.46646267030499352843651622909, 11.75138188651280551331487761974, 12.62466625153211681642075977432, 13.95127112514186933297377284137