# Properties

 Label 2-3150-5.4-c1-0-33 Degree $2$ Conductor $3150$ Sign $-0.894 + 0.447i$ Analytic cond. $25.1528$ Root an. cond. $5.01526$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − i·2-s − 4-s + i·7-s + i·8-s − 3·11-s − 2i·13-s + 14-s + 16-s − 3i·17-s + 7·19-s + 3i·22-s − 2·26-s − i·28-s − 6·29-s − 4·31-s − i·32-s + ⋯
 L(s)  = 1 − 0.707i·2-s − 0.5·4-s + 0.377i·7-s + 0.353i·8-s − 0.904·11-s − 0.554i·13-s + 0.267·14-s + 0.250·16-s − 0.727i·17-s + 1.60·19-s + 0.639i·22-s − 0.392·26-s − 0.188i·28-s − 1.11·29-s − 0.718·31-s − 0.176i·32-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$3150$$    =    $$2 \cdot 3^{2} \cdot 5^{2} \cdot 7$$ Sign: $-0.894 + 0.447i$ Analytic conductor: $$25.1528$$ Root analytic conductor: $$5.01526$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{3150} (2899, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 3150,\ (\ :1/2),\ -0.894 + 0.447i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.9810938421$$ $$L(\frac12)$$ $$\approx$$ $$0.9810938421$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + iT$$
3 $$1$$
5 $$1$$
7 $$1 - iT$$
good11 $$1 + 3T + 11T^{2}$$
13 $$1 + 2iT - 13T^{2}$$
17 $$1 + 3iT - 17T^{2}$$
19 $$1 - 7T + 19T^{2}$$
23 $$1 - 23T^{2}$$
29 $$1 + 6T + 29T^{2}$$
31 $$1 + 4T + 31T^{2}$$
37 $$1 - 8iT - 37T^{2}$$
41 $$1 - 9T + 41T^{2}$$
43 $$1 + 8iT - 43T^{2}$$
47 $$1 - 6iT - 47T^{2}$$
53 $$1 + 12iT - 53T^{2}$$
59 $$1 - 12T + 59T^{2}$$
61 $$1 + 10T + 61T^{2}$$
67 $$1 + 7iT - 67T^{2}$$
71 $$1 + 6T + 71T^{2}$$
73 $$1 + 5iT - 73T^{2}$$
79 $$1 + 14T + 79T^{2}$$
83 $$1 + 9iT - 83T^{2}$$
89 $$1 + 15T + 89T^{2}$$
97 $$1 + 10iT - 97T^{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

−8.422744676750707553716049824440, −7.68023298155020637923512157752, −7.07982932957253178256932905795, −5.67219923136630603838785016438, −5.39430893478400508669661365282, −4.44876803856771021257999655318, −3.28397378274944978768731934921, −2.78582662491447571525563785209, −1.65270176310480196297829155039, −0.32523492560136225421712528518, 1.21157222186398183894077341966, 2.55444404414442935285995910406, 3.68302071037248763598897225823, 4.37657507233285635145635599896, 5.46866095553103841793803729044, 5.78708378126927330382522670052, 6.94419783847639042748313663024, 7.50317155980427810880870642999, 8.001046879665573109320359018896, 9.036394571262452559748210518960