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Properties

Label 2-770-385.142-c1-0-17
Degree $2$
Conductor $770$
Sign $0.696 + 0.717i$
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  + (−0.258 − 0.965i)2-s + (−2.59 − 0.695i)3-s + (−0.866 + 0.499i)4-s + (2.04 − 0.898i)5-s + 2.68i·6-s + (1.70 − 2.02i)7-s + (0.707 + 0.707i)8-s + (3.65 + 2.11i)9-s + (−1.39 − 1.74i)10-s + (−1.11 + 3.12i)11-s + (2.59 − 0.695i)12-s + (3.20 + 3.20i)13-s + (−2.39 − 1.12i)14-s + (−5.94 + 0.907i)15-s + (0.500 − 0.866i)16-s + (2.55 + 0.684i)17-s + ⋯
L(s)  = 1  + (−0.183 − 0.683i)2-s + (−1.49 − 0.401i)3-s + (−0.433 + 0.249i)4-s + (0.915 − 0.401i)5-s + 1.09i·6-s + (0.645 − 0.763i)7-s + (0.249 + 0.249i)8-s + (1.21 + 0.704i)9-s + (−0.441 − 0.551i)10-s + (−0.335 + 0.941i)11-s + (0.749 − 0.200i)12-s + (0.887 + 0.887i)13-s + (−0.639 − 0.301i)14-s + (−1.53 + 0.234i)15-s + (0.125 − 0.216i)16-s + (0.619 + 0.166i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.696 + 0.717i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.696 + 0.717i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.696 + 0.717i$
Analytic conductor: \(6.14848\)
Root analytic conductor: \(2.47961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{770} (527, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 770,\ (\ :1/2),\ 0.696 + 0.717i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.947631 - 0.400530i\)
\(L(\frac12)\) \(\approx\) \(0.947631 - 0.400530i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 + 0.965i)T \)
5 \( 1 + (-2.04 + 0.898i)T \)
7 \( 1 + (-1.70 + 2.02i)T \)
11 \( 1 + (1.11 - 3.12i)T \)
good3 \( 1 + (2.59 + 0.695i)T + (2.59 + 1.5i)T^{2} \)
13 \( 1 + (-3.20 - 3.20i)T + 13iT^{2} \)
17 \( 1 + (-2.55 - 0.684i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (2.02 - 3.49i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.69 - 6.32i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + 2.16T + 29T^{2} \)
31 \( 1 + (-4.73 - 8.20i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-6.06 + 1.62i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + 6.53iT - 41T^{2} \)
43 \( 1 + (7.25 + 7.25i)T + 43iT^{2} \)
47 \( 1 + (-4.70 + 1.26i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-6.49 - 1.74i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (3.24 - 1.87i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.95 + 1.12i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.801 + 2.98i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 5.47T + 71T^{2} \)
73 \( 1 + (1.19 - 4.47i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-2.19 + 3.79i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.64 - 5.64i)T + 83iT^{2} \)
89 \( 1 + (-15.0 - 8.69i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.74 - 5.74i)T + 97iT^{2} \)
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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

−10.49316419422297469030248852484, −9.701389522548578023657867839841, −8.637829563124265312458807386763, −7.48629442800851206304017365390, −6.64083215183300069652020974922, −5.60884502688908328346971074456, −4.93639478394188434416424222817, −3.92576843399449579228242534911, −1.83390191518255775579152250074, −1.17389379280969914704584400941, 0.858643840338535203350751761313, 2.80143179701069187327469796665, 4.59042288922284011197313796304, 5.36536484485346184359974267290, 6.08367682940237990571469054306, 6.36889814978436395361630455210, 7.86699077994911173764196451831, 8.689388539424225926807375782349, 9.699134179998772638539811032376, 10.55642247671882711458970523403

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