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Properties

Label 2-770-11.4-c1-0-19
Degree $2$
Conductor $770$
Sign $0.974 + 0.222i$
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 + (0.636 − 1.95i)3-s + (0.309 + 0.951i)4-s + (−0.809 + 0.587i)5-s + (1.66 − 1.21i)6-s + (0.309 + 0.951i)7-s + (−0.309 + 0.951i)8-s + (−1.00 − 0.728i)9-s − 10-s + (1.76 − 2.80i)11-s + 2.05·12-s + (1.83 + 1.33i)13-s + (−0.309 + 0.951i)14-s + (0.636 + 1.95i)15-s + (−0.809 + 0.587i)16-s + (6.02 − 4.37i)17-s + ⋯
L(s)  = 1  + (0.572 + 0.415i)2-s + (0.367 − 1.13i)3-s + (0.154 + 0.475i)4-s + (−0.361 + 0.262i)5-s + (0.680 − 0.494i)6-s + (0.116 + 0.359i)7-s + (−0.109 + 0.336i)8-s + (−0.334 − 0.242i)9-s − 0.316·10-s + (0.533 − 0.845i)11-s + 0.594·12-s + (0.509 + 0.370i)13-s + (−0.0825 + 0.254i)14-s + (0.164 + 0.505i)15-s + (−0.202 + 0.146i)16-s + (1.46 − 1.06i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 + 0.222i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.974 + 0.222i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.974 + 0.222i$
Analytic conductor: \(6.14848\)
Root analytic conductor: \(2.47961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{770} (631, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 770,\ (\ :1/2),\ 0.974 + 0.222i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.43928 - 0.275268i\)
\(L(\frac12)\) \(\approx\) \(2.43928 - 0.275268i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.76 + 2.80i)T \)
good3 \( 1 + (-0.636 + 1.95i)T + (-2.42 - 1.76i)T^{2} \)
13 \( 1 + (-1.83 - 1.33i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-6.02 + 4.37i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (0.263 - 0.810i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 0.651T + 23T^{2} \)
29 \( 1 + (-0.731 - 2.25i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-0.117 - 0.0852i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-0.719 - 2.21i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-1.43 + 4.41i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 11.7T + 43T^{2} \)
47 \( 1 + (1.63 - 5.01i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-0.778 - 0.565i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-0.322 - 0.991i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-2.90 + 2.10i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 0.430T + 67T^{2} \)
71 \( 1 + (8.60 - 6.25i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (0.359 + 1.10i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (8.66 + 6.29i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (0.876 - 0.636i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 3.73T + 89T^{2} \)
97 \( 1 + (-1.19 - 0.869i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.37954226136105479276504155384, −9.101443087779677228195495379607, −8.287969439502877016937625397914, −7.60966822481755897057051634272, −6.81962666384628215005225917266, −6.07357093517167746356584388355, −5.02902471673842524261356104274, −3.64546807447426818734165469203, −2.78325421691987277707480550126, −1.29480050600874330402203066847, 1.44378071889513271861018807048, 3.20601959485726719015000418453, 3.91415657331738077733774303171, 4.60845702990597764177251272666, 5.58494055469725124045888876779, 6.77133132469652186341804092548, 7.925517119977615321370450142259, 8.798813393630803489521567397333, 9.883058693473211964613771926908, 10.15852561013040374087661946249

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