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Properties

Label 2-770-35.13-c1-0-2
Degree $2$
Conductor $770$
Sign $-0.994 - 0.108i$
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 + (−2.02 + 2.02i)3-s − 1.00i·4-s + (−0.515 − 2.17i)5-s − 2.86i·6-s + (1.43 − 2.22i)7-s + (0.707 + 0.707i)8-s − 5.22i·9-s + (1.90 + 1.17i)10-s − 11-s + (2.02 + 2.02i)12-s + (−0.426 + 0.426i)13-s + (0.558 + 2.58i)14-s + (5.45 + 3.36i)15-s − 1.00·16-s + (3.58 + 3.58i)17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (−1.17 + 1.17i)3-s − 0.500i·4-s + (−0.230 − 0.973i)5-s − 1.17i·6-s + (0.541 − 0.840i)7-s + (0.250 + 0.250i)8-s − 1.74i·9-s + (0.601 + 0.371i)10-s − 0.301·11-s + (0.585 + 0.585i)12-s + (−0.118 + 0.118i)13-s + (0.149 + 0.691i)14-s + (1.40 + 0.869i)15-s − 0.250·16-s + (0.870 + 0.870i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 - 0.108i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 - 0.108i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.994 - 0.108i$
Analytic conductor: \(6.14848\)
Root analytic conductor: \(2.47961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{770} (573, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 770,\ (\ :1/2),\ -0.994 - 0.108i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0156964 + 0.289613i\)
\(L(\frac12)\) \(\approx\) \(0.0156964 + 0.289613i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.515 + 2.17i)T \)
7 \( 1 + (-1.43 + 2.22i)T \)
11 \( 1 + T \)
good3 \( 1 + (2.02 - 2.02i)T - 3iT^{2} \)
13 \( 1 + (0.426 - 0.426i)T - 13iT^{2} \)
17 \( 1 + (-3.58 - 3.58i)T + 17iT^{2} \)
19 \( 1 + 8.41T + 19T^{2} \)
23 \( 1 + (-1.54 - 1.54i)T + 23iT^{2} \)
29 \( 1 - 2.15iT - 29T^{2} \)
31 \( 1 - 8.89iT - 31T^{2} \)
37 \( 1 + (2.82 - 2.82i)T - 37iT^{2} \)
41 \( 1 + 12.0iT - 41T^{2} \)
43 \( 1 + (-7.79 - 7.79i)T + 43iT^{2} \)
47 \( 1 + (-5.76 - 5.76i)T + 47iT^{2} \)
53 \( 1 + (7.26 + 7.26i)T + 53iT^{2} \)
59 \( 1 + 5.29T + 59T^{2} \)
61 \( 1 - 5.58iT - 61T^{2} \)
67 \( 1 + (2.95 - 2.95i)T - 67iT^{2} \)
71 \( 1 + 10.8T + 71T^{2} \)
73 \( 1 + (6.37 - 6.37i)T - 73iT^{2} \)
79 \( 1 - 12.8iT - 79T^{2} \)
83 \( 1 + (5.16 - 5.16i)T - 83iT^{2} \)
89 \( 1 - 5.16T + 89T^{2} \)
97 \( 1 + (-1.87 - 1.87i)T + 97iT^{2} \)
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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

−10.60160666903367881666417931230, −10.08753601918131687943739499253, −9.006274470887550368079530560135, −8.336559969975538360194336210838, −7.29192895980962015521906630144, −6.15616421151011835848315963616, −5.31799468754136796431790830846, −4.58704777033425764563619244848, −3.89206393608693456874912497904, −1.28905225526314642993535761546, 0.21777853250784532910906188044, 1.92185819133673532009287586773, 2.76009257059392704854963146336, 4.50734524621536196149631935894, 5.78035345522113216393408054769, 6.37612731778755738432108383075, 7.47306700084403954299810687971, 7.87550922360946874802312409225, 9.037474381329349921152387876469, 10.28957975747649181525252652126

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