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Properties

Label 2-770-11.3-c1-0-12
Degree $2$
Conductor $770$
Sign $-0.0834 + 0.996i$
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  + (−0.809 + 0.587i)2-s + (−0.897 − 2.76i)3-s + (0.309 − 0.951i)4-s + (0.809 + 0.587i)5-s + (2.34 + 1.70i)6-s + (−0.309 + 0.951i)7-s + (0.309 + 0.951i)8-s + (−4.39 + 3.19i)9-s − 10-s + (3.20 − 0.834i)11-s − 2.90·12-s + (5.26 − 3.82i)13-s + (−0.309 − 0.951i)14-s + (0.897 − 2.76i)15-s + (−0.809 − 0.587i)16-s + (−5.72 − 4.16i)17-s + ⋯
L(s)  = 1  + (−0.572 + 0.415i)2-s + (−0.518 − 1.59i)3-s + (0.154 − 0.475i)4-s + (0.361 + 0.262i)5-s + (0.959 + 0.697i)6-s + (−0.116 + 0.359i)7-s + (0.109 + 0.336i)8-s + (−1.46 + 1.06i)9-s − 0.316·10-s + (0.967 − 0.251i)11-s − 0.838·12-s + (1.46 − 1.06i)13-s + (−0.0825 − 0.254i)14-s + (0.231 − 0.713i)15-s + (−0.202 − 0.146i)16-s + (−1.38 − 1.00i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0834 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0834 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.0834 + 0.996i$
Analytic conductor: \(6.14848\)
Root analytic conductor: \(2.47961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{770} (421, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 770,\ (\ :1/2),\ -0.0834 + 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.689289 - 0.749408i\)
\(L(\frac12)\) \(\approx\) \(0.689289 - 0.749408i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.809 - 0.587i)T \)
5 \( 1 + (-0.809 - 0.587i)T \)
7 \( 1 + (0.309 - 0.951i)T \)
11 \( 1 + (-3.20 + 0.834i)T \)
good3 \( 1 + (0.897 + 2.76i)T + (-2.42 + 1.76i)T^{2} \)
13 \( 1 + (-5.26 + 3.82i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (5.72 + 4.16i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-1.22 - 3.78i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 9.39T + 23T^{2} \)
29 \( 1 + (-0.364 + 1.12i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-1.03 + 0.754i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (0.311 - 0.958i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (1.67 + 5.16i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 11.2T + 43T^{2} \)
47 \( 1 + (3.52 + 10.8i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-6.27 + 4.55i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (1.74 - 5.36i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (5.12 + 3.72i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 3.88T + 67T^{2} \)
71 \( 1 + (-0.210 - 0.152i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-1.49 + 4.58i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (11.6 - 8.43i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (3.71 + 2.70i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 1.43T + 89T^{2} \)
97 \( 1 + (-6.96 + 5.05i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.09545726717059024730867970597, −8.813576617450974642072911164659, −8.471813132459579811983518696148, −7.23339759200041568945620949813, −6.69574276707578067652587797278, −6.04144837090426912314193212802, −5.22788970587006598197869271984, −3.19893877018372271550424117283, −1.82721945723890903553457146202, −0.75720694913220031175536276100, 1.39526451869873737992727958027, 3.25415611872368092759012300156, 4.20553442194120441840297272700, 4.79110943889755773086784060614, 6.24428482630918018422426726832, 6.82087259509255607011068455643, 8.716326360316182902471014195541, 8.951944671825750404417444692389, 9.655123032579849817743315646897, 10.59711996766339975694049427026

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