Properties

Label 2-770-11.9-c1-0-17
Degree $2$
Conductor $770$
Sign $0.993 + 0.117i$
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  + (−0.309 + 0.951i)2-s + (2.31 − 1.68i)3-s + (−0.809 − 0.587i)4-s + (0.309 + 0.951i)5-s + (0.885 + 2.72i)6-s + (−0.809 − 0.587i)7-s + (0.809 − 0.587i)8-s + (1.61 − 4.95i)9-s − 0.999·10-s + (3.27 − 0.497i)11-s − 2.86·12-s + (−1.74 + 5.36i)13-s + (0.809 − 0.587i)14-s + (2.31 + 1.68i)15-s + (0.309 + 0.951i)16-s + (−0.719 − 2.21i)17-s + ⋯
L(s)  = 1  + (−0.218 + 0.672i)2-s + (1.33 − 0.972i)3-s + (−0.404 − 0.293i)4-s + (0.138 + 0.425i)5-s + (0.361 + 1.11i)6-s + (−0.305 − 0.222i)7-s + (0.286 − 0.207i)8-s + (0.536 − 1.65i)9-s − 0.316·10-s + (0.988 − 0.150i)11-s − 0.827·12-s + (−0.483 + 1.48i)13-s + (0.216 − 0.157i)14-s + (0.598 + 0.434i)15-s + (0.0772 + 0.237i)16-s + (−0.174 − 0.537i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.117i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 + 0.117i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.993 + 0.117i$
Analytic conductor: \(6.14848\)
Root analytic conductor: \(2.47961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{770} (141, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 770,\ (\ :1/2),\ 0.993 + 0.117i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.16711 - 0.127710i\)
\(L(\frac12)\) \(\approx\) \(2.16711 - 0.127710i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.309 - 0.951i)T \)
5 \( 1 + (-0.309 - 0.951i)T \)
7 \( 1 + (0.809 + 0.587i)T \)
11 \( 1 + (-3.27 + 0.497i)T \)
good3 \( 1 + (-2.31 + 1.68i)T + (0.927 - 2.85i)T^{2} \)
13 \( 1 + (1.74 - 5.36i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (0.719 + 2.21i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-6.32 + 4.59i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 7.98T + 23T^{2} \)
29 \( 1 + (2.42 + 1.76i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-1.57 + 4.83i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-1.12 - 0.815i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (5.58 - 4.05i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 11.5T + 43T^{2} \)
47 \( 1 + (-4.16 + 3.02i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-0.180 + 0.554i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (11.1 + 8.12i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-2.41 - 7.44i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 0.0485T + 67T^{2} \)
71 \( 1 + (-1.37 - 4.23i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-0.272 - 0.197i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-0.325 + 1.00i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-3.34 - 10.3i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 18.5T + 89T^{2} \)
97 \( 1 + (1.64 - 5.07i)T + (-78.4 - 57.0i)T^{2} \)
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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

−9.598086965144788638692619074998, −9.387618811834161295741646314949, −8.610835514931095423328292591108, −7.49754349604351223408792924345, −6.92377866504907339646395287095, −6.55372890264992388197787031895, −4.91708431475231162451501240881, −3.61875658777916422540024270491, −2.60025199134365500048133350244, −1.26685381403987682378336485733, 1.51966632161400432708982381953, 3.07034420254592670669583017108, 3.42343635376465216131335927313, 4.64699331502863803707967821482, 5.53657973480798812753523141919, 7.21897427861939112112354172200, 8.210935481239367760004489536033, 8.827102196319078266510178662216, 9.551239864153917145950906725651, 10.05400564881595797990773569233

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