Properties

Label 2-770-7.2-c1-0-8
Degree $2$
Conductor $770$
Sign $-0.253 - 0.967i$
Analytic cond. $6.14848$
Root an. cond. $2.47961$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)2-s + (0.922 + 1.59i)3-s + (−0.499 − 0.866i)4-s + (−0.5 + 0.866i)5-s − 1.84·6-s + (2.59 − 0.506i)7-s + 0.999·8-s + (−0.201 + 0.349i)9-s + (−0.499 − 0.866i)10-s + (−0.5 − 0.866i)11-s + (0.922 − 1.59i)12-s + 4.07·13-s + (−0.859 + 2.50i)14-s − 1.84·15-s + (−0.5 + 0.866i)16-s + (3.51 + 6.09i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.532 + 0.922i)3-s + (−0.249 − 0.433i)4-s + (−0.223 + 0.387i)5-s − 0.753·6-s + (0.981 − 0.191i)7-s + 0.353·8-s + (−0.0672 + 0.116i)9-s + (−0.158 − 0.273i)10-s + (−0.150 − 0.261i)11-s + (0.266 − 0.461i)12-s + 1.12·13-s + (−0.229 + 0.668i)14-s − 0.476·15-s + (−0.125 + 0.216i)16-s + (0.853 + 1.47i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.253 - 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.253 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.253 - 0.967i$
Analytic conductor: \(6.14848\)
Root analytic conductor: \(2.47961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{770} (331, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 770,\ (\ :1/2),\ -0.253 - 0.967i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.03197 + 1.33685i\)
\(L(\frac12)\) \(\approx\) \(1.03197 + 1.33685i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (-2.59 + 0.506i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
good3 \( 1 + (-0.922 - 1.59i)T + (-1.5 + 2.59i)T^{2} \)
13 \( 1 - 4.07T + 13T^{2} \)
17 \( 1 + (-3.51 - 6.09i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.22 + 2.11i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.78 - 4.81i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 0.245T + 29T^{2} \)
31 \( 1 + (2.22 + 3.84i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.79 - 4.84i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 8.26T + 41T^{2} \)
43 \( 1 + 1.56T + 43T^{2} \)
47 \( 1 + (-2.44 + 4.22i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.77 + 4.80i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.15 - 3.73i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.92 - 3.33i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.34 - 2.32i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 10.6T + 71T^{2} \)
73 \( 1 + (8.38 + 14.5i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.25 - 9.09i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 10.8T + 83T^{2} \)
89 \( 1 + (1.23 - 2.14i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 6.25T + 97T^{2} \)
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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

−10.38518349800459894635739896976, −9.739136453909784640324751557067, −8.579908723886539961550378219609, −8.280607132329935425079001239402, −7.29613102277600221770842779115, −6.15600522351451622055389234147, −5.23908467954794309333585065399, −4.06544077304013137284178778275, −3.42213953609200855325507197110, −1.51720719381139141444660881168, 1.09529631868568785232808973671, 2.02037818854087896694741839382, 3.21429221262610674295723559414, 4.51462760603990255935035671783, 5.49241753367894110711763420995, 6.95224872129033583123913347619, 7.76081499787699674310022594681, 8.334129055301956100250769474471, 8.992519471073896349417565037904, 10.11427452286958924959815059081

Graph of the $Z$-function along the critical line