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Properties

Label 2-770-55.43-c1-0-34
Degree $2$
Conductor $770$
Sign $-0.908 + 0.416i$
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 + (−1.40 − 1.40i)3-s + 1.00i·4-s + (−0.177 − 2.22i)5-s − 1.98i·6-s + (−0.707 − 0.707i)7-s + (−0.707 + 0.707i)8-s + 0.958i·9-s + (1.45 − 1.70i)10-s + (−2.05 + 2.60i)11-s + (1.40 − 1.40i)12-s + (2.60 − 2.60i)13-s − 1.00i·14-s + (−2.88 + 3.38i)15-s − 1.00·16-s + (1.44 + 1.44i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (−0.812 − 0.812i)3-s + 0.500i·4-s + (−0.0795 − 0.996i)5-s − 0.812i·6-s + (−0.267 − 0.267i)7-s + (−0.250 + 0.250i)8-s + 0.319i·9-s + (0.458 − 0.538i)10-s + (−0.618 + 0.785i)11-s + (0.406 − 0.406i)12-s + (0.722 − 0.722i)13-s − 0.267i·14-s + (−0.745 + 0.874i)15-s − 0.250·16-s + (0.350 + 0.350i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.908 + 0.416i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.908 + 0.416i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.908 + 0.416i$
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: \((2,\ 770,\ (\ :1/2),\ -0.908 + 0.416i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.127345 - 0.583018i\)
\(L(\frac12)\) \(\approx\) \(0.127345 - 0.583018i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.177 + 2.22i)T \)
7 \( 1 + (0.707 + 0.707i)T \)
11 \( 1 + (2.05 - 2.60i)T \)
good3 \( 1 + (1.40 + 1.40i)T + 3iT^{2} \)
13 \( 1 + (-2.60 + 2.60i)T - 13iT^{2} \)
17 \( 1 + (-1.44 - 1.44i)T + 17iT^{2} \)
19 \( 1 + 2.90T + 19T^{2} \)
23 \( 1 + (5.60 + 5.60i)T + 23iT^{2} \)
29 \( 1 + 7.01T + 29T^{2} \)
31 \( 1 + 2.04T + 31T^{2} \)
37 \( 1 + (2.35 - 2.35i)T - 37iT^{2} \)
41 \( 1 + 1.47iT - 41T^{2} \)
43 \( 1 + (-1.73 + 1.73i)T - 43iT^{2} \)
47 \( 1 + (1.81 - 1.81i)T - 47iT^{2} \)
53 \( 1 + (-3.72 - 3.72i)T + 53iT^{2} \)
59 \( 1 - 8.95iT - 59T^{2} \)
61 \( 1 + 13.4iT - 61T^{2} \)
67 \( 1 + (-9.72 + 9.72i)T - 67iT^{2} \)
71 \( 1 + 9.43T + 71T^{2} \)
73 \( 1 + (-2.09 + 2.09i)T - 73iT^{2} \)
79 \( 1 + 11.1T + 79T^{2} \)
83 \( 1 + (-9.44 + 9.44i)T - 83iT^{2} \)
89 \( 1 + 17.2iT - 89T^{2} \)
97 \( 1 + (2.04 - 2.04i)T - 97iT^{2} \)
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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

−10.03441293917742178407827533023, −8.852462963397301754873942476434, −7.978131879342909551772123586348, −7.29910722810025263652888746597, −6.20469079967096465896714191498, −5.70181668080012735350527579856, −4.69054995436603655321832345470, −3.68285693913123739876436072636, −1.86250530835165804663057027532, −0.26824770660760282970022225996, 2.15415965489036715544777969414, 3.46532992680019845066722188837, 4.11835010828598633132124338866, 5.54371645427545338181343490090, 5.84912652126055817661426515179, 6.95981721418097810765345847352, 8.140513489651231315861543072661, 9.408965397137799023502036114423, 10.10541385605464842488980525537, 10.87575786632563622315073868430

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