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Properties

Label 2-770-35.27-c1-0-4
Degree $2$
Conductor $770$
Sign $-0.306 - 0.951i$
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 + (−2.23 − 2.23i)3-s + 1.00i·4-s + (−0.288 + 2.21i)5-s − 3.15i·6-s + (2.06 − 1.64i)7-s + (−0.707 + 0.707i)8-s + 6.95i·9-s + (−1.77 + 1.36i)10-s − 11-s + (2.23 − 2.23i)12-s + (−2.34 − 2.34i)13-s + (2.62 + 0.298i)14-s + (5.59 − 4.30i)15-s − 1.00·16-s + (−3.89 + 3.89i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (−1.28 − 1.28i)3-s + 0.500i·4-s + (−0.128 + 0.991i)5-s − 1.28i·6-s + (0.782 − 0.622i)7-s + (−0.250 + 0.250i)8-s + 2.31i·9-s + (−0.560 + 0.431i)10-s − 0.301·11-s + (0.644 − 0.644i)12-s + (−0.650 − 0.650i)13-s + (0.702 + 0.0797i)14-s + (1.44 − 1.11i)15-s − 0.250·16-s + (−0.944 + 0.944i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.306 - 0.951i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.306 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.306 - 0.951i$
Analytic conductor: \(6.14848\)
Root analytic conductor: \(2.47961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{770} (727, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 770,\ (\ :1/2),\ -0.306 - 0.951i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.470098 + 0.645115i\)
\(L(\frac12)\) \(\approx\) \(0.470098 + 0.645115i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.288 - 2.21i)T \)
7 \( 1 + (-2.06 + 1.64i)T \)
11 \( 1 + T \)
good3 \( 1 + (2.23 + 2.23i)T + 3iT^{2} \)
13 \( 1 + (2.34 + 2.34i)T + 13iT^{2} \)
17 \( 1 + (3.89 - 3.89i)T - 17iT^{2} \)
19 \( 1 - 1.89T + 19T^{2} \)
23 \( 1 + (0.0242 - 0.0242i)T - 23iT^{2} \)
29 \( 1 - 8.09iT - 29T^{2} \)
31 \( 1 - 8.61iT - 31T^{2} \)
37 \( 1 + (-6.66 - 6.66i)T + 37iT^{2} \)
41 \( 1 - 11.7iT - 41T^{2} \)
43 \( 1 + (1.62 - 1.62i)T - 43iT^{2} \)
47 \( 1 + (5.09 - 5.09i)T - 47iT^{2} \)
53 \( 1 + (-6.27 + 6.27i)T - 53iT^{2} \)
59 \( 1 + 0.608T + 59T^{2} \)
61 \( 1 + 10.2iT - 61T^{2} \)
67 \( 1 + (5.29 + 5.29i)T + 67iT^{2} \)
71 \( 1 + 4.98T + 71T^{2} \)
73 \( 1 + (2.98 + 2.98i)T + 73iT^{2} \)
79 \( 1 - 5.09iT - 79T^{2} \)
83 \( 1 + (-4.36 - 4.36i)T + 83iT^{2} \)
89 \( 1 + 18.5T + 89T^{2} \)
97 \( 1 + (6.57 - 6.57i)T - 97iT^{2} \)
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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

−10.93559847225832138287482990119, −10.15567168127776500600358698350, −8.208305883519819365812380462965, −7.72166319719038835087111233191, −6.84082227139879128816113385963, −6.46518851780603143841320738622, −5.34784709961157689723987633458, −4.62969703471891255988373029860, −2.98524900650220493034114483920, −1.54612769636967410246084955732, 0.40464666933772545793117149156, 2.31396259506591609066668982269, 4.24422027479105688253357965587, 4.44257869920695548544081377815, 5.44571817545022381244733194169, 5.84884226554138200237243524722, 7.36197573648041504723943734290, 8.830310442052059973723597944254, 9.417383212701928273728400650007, 10.13002305025372317807119932101

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