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Properties

Label 2-770-7.4-c1-0-0
Degree $2$
Conductor $770$
Sign $-0.781 - 0.624i$
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 + (−0.132 + 0.229i)3-s + (−0.499 + 0.866i)4-s + (0.5 + 0.866i)5-s + 0.265·6-s + (−1.51 + 2.16i)7-s + 0.999·8-s + (1.46 + 2.53i)9-s + (0.499 − 0.866i)10-s + (0.5 − 0.866i)11-s + (−0.132 − 0.229i)12-s − 6.16·13-s + (2.63 + 0.229i)14-s − 0.265·15-s + (−0.5 − 0.866i)16-s + (−1.38 + 2.39i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.0766 + 0.132i)3-s + (−0.249 + 0.433i)4-s + (0.223 + 0.387i)5-s + 0.108·6-s + (−0.573 + 0.819i)7-s + 0.353·8-s + (0.488 + 0.845i)9-s + (0.158 − 0.273i)10-s + (0.150 − 0.261i)11-s + (−0.0383 − 0.0663i)12-s − 1.71·13-s + (0.704 + 0.0614i)14-s − 0.0685·15-s + (−0.125 − 0.216i)16-s + (−0.335 + 0.581i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.781 - 0.624i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.781 - 0.624i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.781 - 0.624i$
Analytic conductor: \(6.14848\)
Root analytic conductor: \(2.47961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{770} (221, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 770,\ (\ :1/2),\ -0.781 - 0.624i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.120315 + 0.343431i\)
\(L(\frac12)\) \(\approx\) \(0.120315 + 0.343431i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.51 - 2.16i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
good3 \( 1 + (0.132 - 0.229i)T + (-1.5 - 2.59i)T^{2} \)
13 \( 1 + 6.16T + 13T^{2} \)
17 \( 1 + (1.38 - 2.39i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.93 + 5.08i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.50 + 6.06i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 0.503T + 29T^{2} \)
31 \( 1 + (1.09 - 1.90i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.28 + 2.21i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 5.53T + 41T^{2} \)
43 \( 1 - 0.993T + 43T^{2} \)
47 \( 1 + (-5.03 - 8.71i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.58 - 9.67i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.950 + 1.64i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.23 + 3.86i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.23 - 5.60i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 16.2T + 71T^{2} \)
73 \( 1 + (1.91 - 3.32i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-7.69 - 13.3i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 13.3T + 83T^{2} \)
89 \( 1 + (6.14 + 10.6i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 12.7T + 97T^{2} \)
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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

−10.57708083032705367387006207250, −9.888272964572458848616331481049, −9.148604867547349773991252629120, −8.270062759107243158158862057262, −7.23174995562241842282640980985, −6.37158488142574246939578115961, −5.14242970501008380652335085350, −4.25477301908485800987012099886, −2.72919703917046607895273197248, −2.15940336526292392462073156499, 0.19493791885114758123729570319, 1.81709163487065311917562451222, 3.63355572285191973201997019015, 4.59940540871865026433028571570, 5.67682942318558260586968892038, 6.71563718174064170899347727404, 7.25281589955388279413381989683, 8.113884386302396397203768445040, 9.429116807842478751555862357746, 9.718398686374754637840395747704

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