Properties

 Degree 2 Conductor $2 \cdot 3^{2} \cdot 5^{2} \cdot 7$ Sign $-0.296 + 0.955i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

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Dirichlet series

 L(s)  = 1 + (0.707 − 0.707i)2-s − 1.00i·4-s + (0.707 + 0.707i)7-s + (−0.707 − 0.707i)8-s − 3.41i·11-s + (1.43 − 1.43i)13-s + 1.00·14-s − 1.00·16-s + (1.54 − 1.54i)17-s + 2.04i·19-s + (−2.41 − 2.41i)22-s + (1.15 + 1.15i)23-s − 2.03i·26-s + (0.707 − 0.707i)28-s + 8.04·29-s + ⋯
 L(s)  = 1 + (0.499 − 0.499i)2-s − 0.500i·4-s + (0.267 + 0.267i)7-s + (−0.250 − 0.250i)8-s − 1.02i·11-s + (0.399 − 0.399i)13-s + 0.267·14-s − 0.250·16-s + (0.374 − 0.374i)17-s + 0.470i·19-s + (−0.514 − 0.514i)22-s + (0.241 + 0.241i)23-s − 0.399i·26-s + (0.133 − 0.133i)28-s + 1.49·29-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.296 + 0.955i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.296 + 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 $$d$$ = $$2$$ $$N$$ = $$3150$$    =    $$2 \cdot 3^{2} \cdot 5^{2} \cdot 7$$ $$\varepsilon$$ = $-0.296 + 0.955i$ motivic weight = $$1$$ character : $\chi_{3150} (1457, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 3150,\ (\ :1/2),\ -0.296 + 0.955i)$ $L(1)$ $\approx$ $2.367930246$ $L(\frac12)$ $\approx$ $2.367930246$ $L(\frac{3}{2})$ 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 + (-0.707 + 0.707i)T$$
3 $$1$$
5 $$1$$
7 $$1 + (-0.707 - 0.707i)T$$
good11 $$1 + 3.41iT - 11T^{2}$$
13 $$1 + (-1.43 + 1.43i)T - 13iT^{2}$$
17 $$1 + (-1.54 + 1.54i)T - 17iT^{2}$$
19 $$1 - 2.04iT - 19T^{2}$$
23 $$1 + (-1.15 - 1.15i)T + 23iT^{2}$$
29 $$1 - 8.04T + 29T^{2}$$
31 $$1 + 6.37T + 31T^{2}$$
37 $$1 + (0.0498 + 0.0498i)T + 37iT^{2}$$
41 $$1 + 7.94iT - 41T^{2}$$
43 $$1 + (-4.65 + 4.65i)T - 43iT^{2}$$
47 $$1 + (-1.39 + 1.39i)T - 47iT^{2}$$
53 $$1 + (2.97 + 2.97i)T + 53iT^{2}$$
59 $$1 + 4.25T + 59T^{2}$$
61 $$1 + 6.88T + 61T^{2}$$
67 $$1 + (5.84 + 5.84i)T + 67iT^{2}$$
71 $$1 - 2.92iT - 71T^{2}$$
73 $$1 + (-5.97 + 5.97i)T - 73iT^{2}$$
79 $$1 - 11.6iT - 79T^{2}$$
83 $$1 + (-1.52 - 1.52i)T + 83iT^{2}$$
89 $$1 - 5.72T + 89T^{2}$$
97 $$1 + (9.64 + 9.64i)T + 97iT^{2}$$
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\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

−8.562635830035805185500825576818, −7.78059928596646795093285837574, −6.86213890280736658613942733895, −5.87008464224038805233185788107, −5.49801464230711334551686652917, −4.53560309829002885037503376506, −3.55265849299716031694199798624, −2.95552284891903965096566697286, −1.80628484027739147500064729673, −0.65507076169128815026289976476, 1.30617622343175463711703285392, 2.48798244637996470348067801365, 3.50952805597037217430825853028, 4.49830013401196710822783603625, 4.85615262580955335260510234835, 5.99331825399816553681238279242, 6.59394387852837539444900971866, 7.39960954366621599159434687893, 7.937997371252991438517879282689, 8.843058318605898012027711949054