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Properties

Label 2-770-35.27-c1-0-35
Degree $2$
Conductor $770$
Sign $-0.595 + 0.803i$
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 + (−0.602 − 0.602i)3-s + 1.00i·4-s + (−0.490 + 2.18i)5-s − 0.852i·6-s + (−2.60 − 0.485i)7-s + (−0.707 + 0.707i)8-s − 2.27i·9-s + (−1.88 + 1.19i)10-s − 11-s + (0.602 − 0.602i)12-s + (−1.45 − 1.45i)13-s + (−1.49 − 2.18i)14-s + (1.61 − 1.01i)15-s − 1.00·16-s + (−0.392 + 0.392i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (−0.347 − 0.347i)3-s + 0.500i·4-s + (−0.219 + 0.975i)5-s − 0.347i·6-s + (−0.982 − 0.183i)7-s + (−0.250 + 0.250i)8-s − 0.757i·9-s + (−0.597 + 0.378i)10-s − 0.301·11-s + (0.173 − 0.173i)12-s + (−0.404 − 0.404i)13-s + (−0.399 − 0.583i)14-s + (0.415 − 0.263i)15-s − 0.250·16-s + (−0.0952 + 0.0952i)17-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(0.0771366 - 0.153169i\)
\(L(\frac12)\) \(\approx\) \(0.0771366 - 0.153169i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.490 - 2.18i)T \)
7 \( 1 + (2.60 + 0.485i)T \)
11 \( 1 + T \)
good3 \( 1 + (0.602 + 0.602i)T + 3iT^{2} \)
13 \( 1 + (1.45 + 1.45i)T + 13iT^{2} \)
17 \( 1 + (0.392 - 0.392i)T - 17iT^{2} \)
19 \( 1 + 5.59T + 19T^{2} \)
23 \( 1 + (-2.20 + 2.20i)T - 23iT^{2} \)
29 \( 1 + 8.31iT - 29T^{2} \)
31 \( 1 - 0.356iT - 31T^{2} \)
37 \( 1 + (-3.62 - 3.62i)T + 37iT^{2} \)
41 \( 1 + 9.90iT - 41T^{2} \)
43 \( 1 + (3.77 - 3.77i)T - 43iT^{2} \)
47 \( 1 + (4.76 - 4.76i)T - 47iT^{2} \)
53 \( 1 + (8.17 - 8.17i)T - 53iT^{2} \)
59 \( 1 + 8.42T + 59T^{2} \)
61 \( 1 + 2.21iT - 61T^{2} \)
67 \( 1 + (0.880 + 0.880i)T + 67iT^{2} \)
71 \( 1 + 5.71T + 71T^{2} \)
73 \( 1 + (1.48 + 1.48i)T + 73iT^{2} \)
79 \( 1 - 7.65iT - 79T^{2} \)
83 \( 1 + (5.51 + 5.51i)T + 83iT^{2} \)
89 \( 1 - 12.1T + 89T^{2} \)
97 \( 1 + (-2.77 + 2.77i)T - 97iT^{2} \)
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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

−10.11220337666564442241183658302, −9.184919704930252538313672354765, −7.975981941209911517921286235626, −7.18341283161434471143778660692, −6.31905327318387683069360768870, −6.07828614675050027016154077356, −4.50587629599474345364473857375, −3.48680212229523844460174192606, −2.57751707484291969395414872056, −0.07119458781392514786955889045, 1.87808931719049486050057839549, 3.21758303450145898741691298160, 4.41368043128344134171784147114, 5.03097364946777795668651265568, 5.94009743255260792755322412648, 7.00209482812185383874681133356, 8.214047729071721881677807127708, 9.128113124405472137511732682368, 9.852558285830648122981239983635, 10.68530805897223334238091446479

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