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Properties

Label 2-770-7.2-c1-0-12
Degree $2$
Conductor $770$
Sign $0.635 + 0.771i$
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 + (−0.112 − 0.195i)3-s + (−0.499 − 0.866i)4-s + (−0.5 + 0.866i)5-s + 0.225·6-s + (−2.14 + 1.54i)7-s + 0.999·8-s + (1.47 − 2.55i)9-s + (−0.499 − 0.866i)10-s + (−0.5 − 0.866i)11-s + (−0.112 + 0.195i)12-s − 0.878·13-s + (−0.269 − 2.63i)14-s + 0.225·15-s + (−0.5 + 0.866i)16-s + (−2.25 − 3.91i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.0650 − 0.112i)3-s + (−0.249 − 0.433i)4-s + (−0.223 + 0.387i)5-s + 0.0919·6-s + (−0.810 + 0.585i)7-s + 0.353·8-s + (0.491 − 0.851i)9-s + (−0.158 − 0.273i)10-s + (−0.150 − 0.261i)11-s + (−0.0325 + 0.0563i)12-s − 0.243·13-s + (−0.0719 − 0.703i)14-s + 0.0581·15-s + (−0.125 + 0.216i)16-s + (−0.547 − 0.948i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.635 + 0.771i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.635 + 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.635 + 0.771i$
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: \((2,\ 770,\ (\ :1/2),\ 0.635 + 0.771i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.688017 - 0.324706i\)
\(L(\frac12)\) \(\approx\) \(0.688017 - 0.324706i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.14 - 1.54i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
good3 \( 1 + (0.112 + 0.195i)T + (-1.5 + 2.59i)T^{2} \)
13 \( 1 + 0.878T + 13T^{2} \)
17 \( 1 + (2.25 + 3.91i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.86 + 3.22i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.15 - 2.00i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 7.36T + 29T^{2} \)
31 \( 1 + (2.86 + 4.95i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.79 + 3.11i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 2.52T + 41T^{2} \)
43 \( 1 - 1.68T + 43T^{2} \)
47 \( 1 + (-3.72 + 6.45i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.24 + 7.34i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.85 + 8.40i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.62 + 4.55i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.725 + 1.25i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 2.90T + 71T^{2} \)
73 \( 1 + (-0.661 - 1.14i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.93 - 12.0i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 4.89T + 83T^{2} \)
89 \( 1 + (-4.73 + 8.20i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 13.6T + 97T^{2} \)
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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

−9.793765249775649815443035747479, −9.453173068403658274491739986280, −8.538372329979257650878431106955, −7.40888011807106041944760208721, −6.76803311015031112300030046361, −6.04952477884232565980262918492, −4.96611634350439654849973042796, −3.68325808786897709051777374179, −2.52237931212648941637985975169, −0.46638765305618305728479887086, 1.37038229232794234625686086347, 2.77923126943930636878063480524, 4.03213485201054095901509198637, 4.70411914950072045857040661097, 6.06264764414745297023820230638, 7.19641960462246513976047869437, 7.936892192461264626155659531031, 8.833636722438061054146628580165, 9.821290786005750874374889900477, 10.40757111832235330850512316963

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