Properties

Label 2-322-23.6-c1-0-4
Degree $2$
Conductor $322$
Sign $0.875 + 0.482i$
Analytic cond. $2.57118$
Root an. cond. $1.60349$
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  + (−0.415 + 0.909i)2-s + (−2.45 − 0.721i)3-s + (−0.654 − 0.755i)4-s + (−3.26 + 2.10i)5-s + (1.67 − 1.93i)6-s + (−0.142 + 0.989i)7-s + (0.959 − 0.281i)8-s + (2.99 + 1.92i)9-s + (−0.553 − 3.84i)10-s + (0.793 + 1.73i)11-s + (1.06 + 2.32i)12-s + (−0.713 − 4.96i)13-s + (−0.841 − 0.540i)14-s + (9.54 − 2.80i)15-s + (−0.142 + 0.989i)16-s + (4.11 − 4.75i)17-s + ⋯
L(s)  = 1  + (−0.293 + 0.643i)2-s + (−1.41 − 0.416i)3-s + (−0.327 − 0.377i)4-s + (−1.46 + 0.939i)5-s + (0.684 − 0.789i)6-s + (−0.0537 + 0.374i)7-s + (0.339 − 0.0996i)8-s + (0.997 + 0.640i)9-s + (−0.174 − 1.21i)10-s + (0.239 + 0.523i)11-s + (0.307 + 0.672i)12-s + (−0.197 − 1.37i)13-s + (−0.224 − 0.144i)14-s + (2.46 − 0.723i)15-s + (−0.0355 + 0.247i)16-s + (0.998 − 1.15i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.875 + 0.482i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.875 + 0.482i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(322\)    =    \(2 \cdot 7 \cdot 23\)
Sign: $0.875 + 0.482i$
Analytic conductor: \(2.57118\)
Root analytic conductor: \(1.60349\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{322} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 322,\ (\ :1/2),\ 0.875 + 0.482i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.344785 - 0.0887309i\)
\(L(\frac12)\) \(\approx\) \(0.344785 - 0.0887309i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.415 - 0.909i)T \)
7 \( 1 + (0.142 - 0.989i)T \)
23 \( 1 + (-4.14 - 2.41i)T \)
good3 \( 1 + (2.45 + 0.721i)T + (2.52 + 1.62i)T^{2} \)
5 \( 1 + (3.26 - 2.10i)T + (2.07 - 4.54i)T^{2} \)
11 \( 1 + (-0.793 - 1.73i)T + (-7.20 + 8.31i)T^{2} \)
13 \( 1 + (0.713 + 4.96i)T + (-12.4 + 3.66i)T^{2} \)
17 \( 1 + (-4.11 + 4.75i)T + (-2.41 - 16.8i)T^{2} \)
19 \( 1 + (1.84 + 2.13i)T + (-2.70 + 18.8i)T^{2} \)
29 \( 1 + (1.38 - 1.60i)T + (-4.12 - 28.7i)T^{2} \)
31 \( 1 + (2.75 - 0.808i)T + (26.0 - 16.7i)T^{2} \)
37 \( 1 + (-1.51 - 0.976i)T + (15.3 + 33.6i)T^{2} \)
41 \( 1 + (0.951 - 0.611i)T + (17.0 - 37.2i)T^{2} \)
43 \( 1 + (7.22 + 2.12i)T + (36.1 + 23.2i)T^{2} \)
47 \( 1 - 8.41T + 47T^{2} \)
53 \( 1 + (-1.97 + 13.7i)T + (-50.8 - 14.9i)T^{2} \)
59 \( 1 + (-1.36 - 9.45i)T + (-56.6 + 16.6i)T^{2} \)
61 \( 1 + (-10.4 + 3.07i)T + (51.3 - 32.9i)T^{2} \)
67 \( 1 + (-4.77 + 10.4i)T + (-43.8 - 50.6i)T^{2} \)
71 \( 1 + (1.74 - 3.82i)T + (-46.4 - 53.6i)T^{2} \)
73 \( 1 + (1.24 + 1.43i)T + (-10.3 + 72.2i)T^{2} \)
79 \( 1 + (1.80 + 12.5i)T + (-75.7 + 22.2i)T^{2} \)
83 \( 1 + (8.38 + 5.38i)T + (34.4 + 75.4i)T^{2} \)
89 \( 1 + (14.8 + 4.35i)T + (74.8 + 48.1i)T^{2} \)
97 \( 1 + (-5.45 + 3.50i)T + (40.2 - 88.2i)T^{2} \)
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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

−11.53391640114938052197648201483, −10.82268140684096871185828694637, −9.900015479165536672783500979540, −8.382249955087763578949565726747, −7.17202481072277690673329883646, −7.11899325742899319658833039391, −5.71388689950664445789635610700, −4.87327628819476268778505592214, −3.24084945108201283662138765437, −0.44368005304758956518157148198, 1.04291172126026529596400349932, 3.85751712065039972810756084947, 4.35110419313751614152307958134, 5.53038682750839257106651814710, 6.88099223533847872283112300149, 8.083721906892603555690989040386, 8.948745322107305497722231683952, 10.12497888020610465260878481144, 11.07985103720392274657483475526, 11.55217902797386672592742208654

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