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Properties

Label 2-770-11.9-c1-0-16
Degree $2$
Conductor $770$
Sign $0.395 + 0.918i$
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 + (1.06 − 0.776i)3-s + (−0.809 − 0.587i)4-s + (0.309 + 0.951i)5-s + (−0.408 − 1.25i)6-s + (0.809 + 0.587i)7-s + (−0.809 + 0.587i)8-s + (−0.387 + 1.19i)9-s + 0.999·10-s + (2.07 + 2.59i)11-s − 1.32·12-s + (1.65 − 5.08i)13-s + (0.809 − 0.587i)14-s + (1.06 + 0.776i)15-s + (0.309 + 0.951i)16-s + (−1.99 − 6.13i)17-s + ⋯
L(s)  = 1  + (0.218 − 0.672i)2-s + (0.616 − 0.448i)3-s + (−0.404 − 0.293i)4-s + (0.138 + 0.425i)5-s + (−0.166 − 0.512i)6-s + (0.305 + 0.222i)7-s + (−0.286 + 0.207i)8-s + (−0.129 + 0.398i)9-s + 0.316·10-s + (0.624 + 0.780i)11-s − 0.381·12-s + (0.458 − 1.41i)13-s + (0.216 − 0.157i)14-s + (0.275 + 0.200i)15-s + (0.0772 + 0.237i)16-s + (−0.483 − 1.48i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.395 + 0.918i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.395 + 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.395 + 0.918i$
Analytic conductor: \(6.14848\)
Root analytic conductor: \(2.47961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{770} (141, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 770,\ (\ :1/2),\ 0.395 + 0.918i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.85473 - 1.22130i\)
\(L(\frac12)\) \(\approx\) \(1.85473 - 1.22130i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.07 - 2.59i)T \)
good3 \( 1 + (-1.06 + 0.776i)T + (0.927 - 2.85i)T^{2} \)
13 \( 1 + (-1.65 + 5.08i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (1.99 + 6.13i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-5.19 + 3.77i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 5.51T + 23T^{2} \)
29 \( 1 + (-6.63 - 4.81i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-0.548 + 1.68i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (8.88 + 6.45i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (1.68 - 1.22i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 2.71T + 43T^{2} \)
47 \( 1 + (3.52 - 2.56i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (2.65 - 8.17i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-4.40 - 3.20i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-4.14 - 12.7i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 10.8T + 67T^{2} \)
71 \( 1 + (3.65 + 11.2i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-4.48 - 3.26i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (2.27 - 6.99i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (2.22 + 6.84i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 5.39T + 89T^{2} \)
97 \( 1 + (0.169 - 0.521i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.28985924044873467654113194212, −9.238356459418872421466099902536, −8.671332711441415559455362628783, −7.48395121803512541499415863316, −6.93563842860465710320045580089, −5.42278596737901630516152921741, −4.76757750385339786877922067160, −3.13492660159982194267883639129, −2.65851828071486907745386926496, −1.24685430139700864818950460586, 1.45152467645235386494808124926, 3.37726449251953195815398243368, 4.01437952785798914443095253657, 5.03052815023929093373284427372, 6.24636370094194147107995976506, 6.77412492022026577690523323087, 8.329782095902105776617852714424, 8.527834986555090303136608669662, 9.383905801796632428744147364111, 10.21363746194744358706885477099

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