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Properties

Label 2-770-55.43-c1-0-13
Degree $2$
Conductor $770$
Sign $0.813 - 0.581i$
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  + (0.707 + 0.707i)2-s + (−0.668 − 0.668i)3-s + 1.00i·4-s + (−2.17 + 0.508i)5-s − 0.945i·6-s + (−0.707 − 0.707i)7-s + (−0.707 + 0.707i)8-s − 2.10i·9-s + (−1.89 − 1.17i)10-s + (2.68 + 1.94i)11-s + (0.668 − 0.668i)12-s + (1.94 − 1.94i)13-s − 1.00i·14-s + (1.79 + 1.11i)15-s − 1.00·16-s + (3.61 + 3.61i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (−0.386 − 0.386i)3-s + 0.500i·4-s + (−0.973 + 0.227i)5-s − 0.386i·6-s + (−0.267 − 0.267i)7-s + (−0.250 + 0.250i)8-s − 0.701i·9-s + (−0.600 − 0.373i)10-s + (0.809 + 0.586i)11-s + (0.193 − 0.193i)12-s + (0.539 − 0.539i)13-s − 0.267i·14-s + (0.463 + 0.288i)15-s − 0.250·16-s + (0.875 + 0.875i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.813 - 0.581i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.813 - 0.581i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.813 - 0.581i$
Analytic conductor: \(6.14848\)
Root analytic conductor: \(2.47961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{770} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 770,\ (\ :1/2),\ 0.813 - 0.581i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.47179 + 0.471456i\)
\(L(\frac12)\) \(\approx\) \(1.47179 + 0.471456i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 + (2.17 - 0.508i)T \)
7 \( 1 + (0.707 + 0.707i)T \)
11 \( 1 + (-2.68 - 1.94i)T \)
good3 \( 1 + (0.668 + 0.668i)T + 3iT^{2} \)
13 \( 1 + (-1.94 + 1.94i)T - 13iT^{2} \)
17 \( 1 + (-3.61 - 3.61i)T + 17iT^{2} \)
19 \( 1 - 3.79T + 19T^{2} \)
23 \( 1 + (-6.28 - 6.28i)T + 23iT^{2} \)
29 \( 1 - 4.14T + 29T^{2} \)
31 \( 1 + 5.10T + 31T^{2} \)
37 \( 1 + (6.35 - 6.35i)T - 37iT^{2} \)
41 \( 1 + 5.80iT - 41T^{2} \)
43 \( 1 + (-8.15 + 8.15i)T - 43iT^{2} \)
47 \( 1 + (0.337 - 0.337i)T - 47iT^{2} \)
53 \( 1 + (-5.85 - 5.85i)T + 53iT^{2} \)
59 \( 1 + 13.4iT - 59T^{2} \)
61 \( 1 - 0.995iT - 61T^{2} \)
67 \( 1 + (2.69 - 2.69i)T - 67iT^{2} \)
71 \( 1 - 4.97T + 71T^{2} \)
73 \( 1 + (3.48 - 3.48i)T - 73iT^{2} \)
79 \( 1 - 8.68T + 79T^{2} \)
83 \( 1 + (-8.86 + 8.86i)T - 83iT^{2} \)
89 \( 1 - 6.56iT - 89T^{2} \)
97 \( 1 + (5.10 - 5.10i)T - 97iT^{2} \)
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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

−10.58478380387359254931908952354, −9.426023976121355489357976648161, −8.535436902300272935665021970349, −7.39096646577532464220216436877, −7.06630178280751591847659616752, −6.07408889243850105835889403309, −5.14848095985685918246567442802, −3.72804018725637633893927528722, −3.42225067501817571966675454875, −1.10334643480776330252216520403, 0.967983795942120570987465177448, 2.84697849604628933416068066010, 3.79874673630529532898388085671, 4.73263797053508916791090762594, 5.49436650251305330570497191693, 6.63086304505735771950413127440, 7.60572666508326490927874399262, 8.747544895690562266942709216620, 9.383916174813688607436218511978, 10.56934510175579039798841535281

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