# Properties

 Degree 2 Conductor $2 \cdot 3^{2} \cdot 5^{2} \cdot 7$ Sign $0.309 + 0.950i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 − i·2-s − 4-s + (−2.12 − 1.58i)7-s + i·8-s − 1.41i·11-s + 3.16i·13-s + (−1.58 + 2.12i)14-s + 16-s + 4.47·17-s − 1.41·22-s + 6i·23-s + 3.16·26-s + (2.12 + 1.58i)28-s − 2.82i·29-s − i·32-s + ⋯
 L(s)  = 1 − 0.707i·2-s − 0.5·4-s + (−0.801 − 0.597i)7-s + 0.353i·8-s − 0.426i·11-s + 0.877i·13-s + (−0.422 + 0.566i)14-s + 0.250·16-s + 1.08·17-s − 0.301·22-s + 1.25i·23-s + 0.620·26-s + (0.400 + 0.298i)28-s − 0.525i·29-s − 0.176i·32-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.309 + 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.309 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$3150$$    =    $$2 \cdot 3^{2} \cdot 5^{2} \cdot 7$$ $$\varepsilon$$ = $0.309 + 0.950i$ motivic weight = $$1$$ character : $\chi_{3150} (251, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 3150,\ (\ :1/2),\ 0.309 + 0.950i)$ $L(1)$ $\approx$ $1.471354910$ $L(\frac12)$ $\approx$ $1.471354910$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3,\;5,\;7\}$,$$F_p(T)$$ is a polynomial of degree 2. If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1 + iT$$
3 $$1$$
5 $$1$$
7 $$1 + (2.12 + 1.58i)T$$
good11 $$1 + 1.41iT - 11T^{2}$$
13 $$1 - 3.16iT - 13T^{2}$$
17 $$1 - 4.47T + 17T^{2}$$
19 $$1 - 19T^{2}$$
23 $$1 - 6iT - 23T^{2}$$
29 $$1 + 2.82iT - 29T^{2}$$
31 $$1 - 31T^{2}$$
37 $$1 - 4.24T + 37T^{2}$$
41 $$1 + 9.48T + 41T^{2}$$
43 $$1 - 8.48T + 43T^{2}$$
47 $$1 - 4.47T + 47T^{2}$$
53 $$1 - 6iT - 53T^{2}$$
59 $$1 - 9.48T + 59T^{2}$$
61 $$1 + 13.4iT - 61T^{2}$$
67 $$1 + 67T^{2}$$
71 $$1 + 5.65iT - 71T^{2}$$
73 $$1 + 6.32iT - 73T^{2}$$
79 $$1 - 4T + 79T^{2}$$
83 $$1 - 8.94T + 83T^{2}$$
89 $$1 + 9.48T + 89T^{2}$$
97 $$1 + 12.6iT - 97T^{2}$$
show less
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}

## Imaginary part of the first few zeros on the critical line

−8.683707661707473615537797972849, −7.79189375727999777484225087868, −7.12070260282011359611503176203, −6.18897252039217780966329054809, −5.47922098466323583839562754438, −4.40454637481560450660674396891, −3.65387939743807060257785817763, −3.01818952441766136096994608524, −1.79810788007788625237553595835, −0.67798866511102393119037935785, 0.799780946583711719781980224577, 2.43559521041802132733773474054, 3.28885593833010005628792800811, 4.22822934998891776904029256027, 5.30540466022121611294662912513, 5.73779314262574789108775621242, 6.62994323415416782856492017483, 7.23339823549306703299757511009, 8.135565099030279020115821542908, 8.660132650359010560573903910052