Properties

Label 2-770-11.5-c1-0-11
Degree $2$
Conductor $770$
Sign $0.504 + 0.863i$
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  + (0.309 + 0.951i)2-s + (−2.57 − 1.87i)3-s + (−0.809 + 0.587i)4-s + (−0.309 + 0.951i)5-s + (0.985 − 3.03i)6-s + (0.809 − 0.587i)7-s + (−0.809 − 0.587i)8-s + (2.21 + 6.81i)9-s − 0.999·10-s + (−0.853 − 3.20i)11-s + 3.18·12-s + (1.75 + 5.41i)13-s + (0.809 + 0.587i)14-s + (2.57 − 1.87i)15-s + (0.309 − 0.951i)16-s + (−0.231 + 0.712i)17-s + ⋯
L(s)  = 1  + (0.218 + 0.672i)2-s + (−1.48 − 1.08i)3-s + (−0.404 + 0.293i)4-s + (−0.138 + 0.425i)5-s + (0.402 − 1.23i)6-s + (0.305 − 0.222i)7-s + (−0.286 − 0.207i)8-s + (0.737 + 2.27i)9-s − 0.316·10-s + (−0.257 − 0.966i)11-s + 0.920·12-s + (0.487 + 1.50i)13-s + (0.216 + 0.157i)14-s + (0.665 − 0.483i)15-s + (0.0772 − 0.237i)16-s + (−0.0561 + 0.172i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.504 + 0.863i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.504 + 0.863i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.504 + 0.863i$
Analytic conductor: \(6.14848\)
Root analytic conductor: \(2.47961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{770} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 770,\ (\ :1/2),\ 0.504 + 0.863i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.637493 - 0.365934i\)
\(L(\frac12)\) \(\approx\) \(0.637493 - 0.365934i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.309 - 0.951i)T \)
5 \( 1 + (0.309 - 0.951i)T \)
7 \( 1 + (-0.809 + 0.587i)T \)
11 \( 1 + (0.853 + 3.20i)T \)
good3 \( 1 + (2.57 + 1.87i)T + (0.927 + 2.85i)T^{2} \)
13 \( 1 + (-1.75 - 5.41i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (0.231 - 0.712i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (5.49 + 3.99i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 - 3.94T + 23T^{2} \)
29 \( 1 + (-4.64 + 3.37i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (2.19 + 6.76i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-9.38 + 6.82i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (9.25 + 6.72i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 2.33T + 43T^{2} \)
47 \( 1 + (-4.04 - 2.93i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (3.32 + 10.2i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (4.23 - 3.07i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-2.37 + 7.30i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 7.39T + 67T^{2} \)
71 \( 1 + (0.177 - 0.547i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-10.1 + 7.38i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-2.11 - 6.50i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-3.25 + 10.0i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 4.80T + 89T^{2} \)
97 \( 1 + (-5.02 - 15.4i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.63848580385445820798602715778, −9.111540330435241317610538658872, −8.118998221901030917183151342441, −7.27944121395768317819900979541, −6.47266654278082685180889838473, −6.14163904170846283956133287035, −5.01411933798571462132119770801, −4.09822508884009536744520519174, −2.16729072844583671602957050666, −0.49777201931125311972629229647, 1.16428656433538335853358396575, 3.14778967860981036551474528064, 4.36180487659280945661469489325, 4.92666281309662767112058888033, 5.65977828535680293074612655297, 6.59368081883160219496626945167, 8.094008172179993648356928576058, 9.064867370105323015713064724018, 10.09872598754522171195762985113, 10.48587661040202314734459480617

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