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Properties

Label 2-770-385.54-c1-0-21
Degree $2$
Conductor $770$
Sign $0.890 + 0.454i$
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  + (−0.5 + 0.866i)2-s + (−0.987 − 1.71i)3-s + (−0.499 − 0.866i)4-s + (1.00 + 1.99i)5-s + 1.97·6-s + (1.58 − 2.12i)7-s + 0.999·8-s + (−0.451 + 0.782i)9-s + (−2.23 − 0.127i)10-s + (1.14 + 3.11i)11-s + (−0.987 + 1.71i)12-s − 4.36i·13-s + (1.04 + 2.42i)14-s + (2.42 − 3.69i)15-s + (−0.5 + 0.866i)16-s + (−0.154 + 0.0893i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.570 − 0.987i)3-s + (−0.249 − 0.433i)4-s + (0.449 + 0.893i)5-s + 0.806·6-s + (0.597 − 0.802i)7-s + 0.353·8-s + (−0.150 + 0.260i)9-s + (−0.705 − 0.0404i)10-s + (0.344 + 0.938i)11-s + (−0.285 + 0.493i)12-s − 1.21i·13-s + (0.280 + 0.649i)14-s + (0.625 − 0.953i)15-s + (−0.125 + 0.216i)16-s + (−0.0375 + 0.0216i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.890 + 0.454i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.890 + 0.454i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.890 + 0.454i$
Analytic conductor: \(6.14848\)
Root analytic conductor: \(2.47961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{770} (439, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 770,\ (\ :1/2),\ 0.890 + 0.454i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.16686 - 0.280478i\)
\(L(\frac12)\) \(\approx\) \(1.16686 - 0.280478i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 + (-1.00 - 1.99i)T \)
7 \( 1 + (-1.58 + 2.12i)T \)
11 \( 1 + (-1.14 - 3.11i)T \)
good3 \( 1 + (0.987 + 1.71i)T + (-1.5 + 2.59i)T^{2} \)
13 \( 1 + 4.36iT - 13T^{2} \)
17 \( 1 + (0.154 - 0.0893i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.57 - 2.73i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-6.01 - 3.47i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 7.86iT - 29T^{2} \)
31 \( 1 + (-6.35 + 3.67i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.87 + 1.08i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 7.35T + 41T^{2} \)
43 \( 1 + 1.81T + 43T^{2} \)
47 \( 1 + (-1.99 + 3.46i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-10.2 + 5.91i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.30 - 3.63i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7.80 + 13.5i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.07 + 2.92i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 7.02T + 71T^{2} \)
73 \( 1 + (4.65 - 2.68i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.13 - 2.96i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 2.33iT - 83T^{2} \)
89 \( 1 + (5.10 + 2.94i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 2.52T + 97T^{2} \)
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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

−10.18389404528317147738822574333, −9.591063076098597251367125767184, −8.092277924120831300964503856447, −7.50240825916811640419614050222, −6.86474240067768033556122067597, −6.13324094238148161377470579299, −5.22655959885012323303488937505, −3.87786608780479268274838916781, −2.16009791808413484164307957248, −0.898162557416265785265574205220, 1.25039975205816299294048533538, 2.64318389229700697882582147667, 4.20058731647363194881973899436, 4.84540584513382121753698818071, 5.61572572575601249279651864504, 6.78988634501766646130314668788, 8.378900175639631584690665151420, 8.955257735179269652885566786066, 9.373696592440334845396634145877, 10.54032328127146595129303374294

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