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Properties

Label 2-770-11.5-c1-0-19
Degree $2$
Conductor $770$
Sign $-0.803 + 0.595i$
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 + (0.756 + 0.549i)3-s + (−0.809 + 0.587i)4-s + (0.309 − 0.951i)5-s + (0.289 − 0.889i)6-s + (−0.809 + 0.587i)7-s + (0.809 + 0.587i)8-s + (−0.656 − 2.02i)9-s − 0.999·10-s + (−3.09 + 1.19i)11-s − 0.935·12-s + (−0.776 − 2.39i)13-s + (0.809 + 0.587i)14-s + (0.756 − 0.549i)15-s + (0.309 − 0.951i)16-s + (1.44 − 4.45i)17-s + ⋯
L(s)  = 1  + (−0.218 − 0.672i)2-s + (0.437 + 0.317i)3-s + (−0.404 + 0.293i)4-s + (0.138 − 0.425i)5-s + (0.118 − 0.363i)6-s + (−0.305 + 0.222i)7-s + (0.286 + 0.207i)8-s + (−0.218 − 0.673i)9-s − 0.316·10-s + (−0.932 + 0.361i)11-s − 0.270·12-s + (−0.215 − 0.663i)13-s + (0.216 + 0.157i)14-s + (0.195 − 0.142i)15-s + (0.0772 − 0.237i)16-s + (0.351 − 1.08i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.803 + 0.595i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.803 + 0.595i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.803 + 0.595i$
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: \((2,\ 770,\ (\ :1/2),\ -0.803 + 0.595i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.294703 - 0.891881i\)
\(L(\frac12)\) \(\approx\) \(0.294703 - 0.891881i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.09 - 1.19i)T \)
good3 \( 1 + (-0.756 - 0.549i)T + (0.927 + 2.85i)T^{2} \)
13 \( 1 + (0.776 + 2.39i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-1.44 + 4.45i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (3.28 + 2.38i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 3.46T + 23T^{2} \)
29 \( 1 + (-6.03 + 4.38i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-0.477 - 1.47i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-3.98 + 2.89i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (1.33 + 0.970i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 7.97T + 43T^{2} \)
47 \( 1 + (1.43 + 1.04i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-2.20 - 6.79i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-0.345 + 0.250i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-1.90 + 5.85i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 5.99T + 67T^{2} \)
71 \( 1 + (-2.02 + 6.21i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-0.692 + 0.502i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (3.04 + 9.37i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (2.52 - 7.75i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 1.00T + 89T^{2} \)
97 \( 1 + (-3.50 - 10.7i)T + (-78.4 + 57.0i)T^{2} \)
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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

−9.922258577500371708678566155001, −9.303237883653592001288760780049, −8.455455260563629040537888857638, −7.72539271203945132682810645450, −6.43492825462311410057072805066, −5.27812927409755662693071462403, −4.37938766213566695693554250789, −3.13968874353076293368353828585, −2.38512738202390087968333327758, −0.46245590216285114959954023030, 1.85154599040212934216473961993, 3.08496948289144095536127658995, 4.38463470201895469621593544752, 5.54915603088266330856967347625, 6.40539975284242065562029995586, 7.24873625057444482528188408886, 8.232167425529513173513316740579, 8.491726247305838937386314078986, 9.942894705452927750577804708327, 10.35513360399324296487489958184

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