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Properties

Label 2-522-29.25-c1-0-12
Degree $2$
Conductor $522$
Sign $-0.905 - 0.423i$
Analytic cond. $4.16819$
Root an. cond. $2.04161$
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  + (0.222 − 0.974i)2-s + (−0.900 − 0.433i)4-s + (−0.110 + 0.482i)5-s + (−2.82 + 1.36i)7-s + (−0.623 + 0.781i)8-s + (0.445 + 0.214i)10-s + (−0.870 − 1.09i)11-s + (−3.56 − 4.46i)13-s + (0.698 + 3.05i)14-s + (0.623 + 0.781i)16-s − 5.31·17-s + (−4.47 − 2.15i)19-s + (0.308 − 0.386i)20-s + (−1.25 + 0.606i)22-s + (−0.181 − 0.793i)23-s + ⋯
L(s)  = 1  + (0.157 − 0.689i)2-s + (−0.450 − 0.216i)4-s + (−0.0492 + 0.215i)5-s + (−1.06 + 0.514i)7-s + (−0.220 + 0.276i)8-s + (0.140 + 0.0678i)10-s + (−0.262 − 0.329i)11-s + (−0.988 − 1.23i)13-s + (0.186 + 0.817i)14-s + (0.155 + 0.195i)16-s − 1.28·17-s + (−1.02 − 0.494i)19-s + (0.0689 − 0.0864i)20-s + (−0.268 + 0.129i)22-s + (−0.0377 − 0.165i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 522 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.905 - 0.423i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 522 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.905 - 0.423i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(522\)    =    \(2 \cdot 3^{2} \cdot 29\)
Sign: $-0.905 - 0.423i$
Analytic conductor: \(4.16819\)
Root analytic conductor: \(2.04161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{522} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 522,\ (\ :1/2),\ -0.905 - 0.423i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0484768 + 0.218208i\)
\(L(\frac12)\) \(\approx\) \(0.0484768 + 0.218208i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.222 + 0.974i)T \)
3 \( 1 \)
29 \( 1 + (5.38 - 0.0414i)T \)
good5 \( 1 + (0.110 - 0.482i)T + (-4.50 - 2.16i)T^{2} \)
7 \( 1 + (2.82 - 1.36i)T + (4.36 - 5.47i)T^{2} \)
11 \( 1 + (0.870 + 1.09i)T + (-2.44 + 10.7i)T^{2} \)
13 \( 1 + (3.56 + 4.46i)T + (-2.89 + 12.6i)T^{2} \)
17 \( 1 + 5.31T + 17T^{2} \)
19 \( 1 + (4.47 + 2.15i)T + (11.8 + 14.8i)T^{2} \)
23 \( 1 + (0.181 + 0.793i)T + (-20.7 + 9.97i)T^{2} \)
31 \( 1 + (1.41 - 6.20i)T + (-27.9 - 13.4i)T^{2} \)
37 \( 1 + (-5.56 + 6.97i)T + (-8.23 - 36.0i)T^{2} \)
41 \( 1 - 4.01T + 41T^{2} \)
43 \( 1 + (-0.310 - 1.36i)T + (-38.7 + 18.6i)T^{2} \)
47 \( 1 + (-6.42 - 8.05i)T + (-10.4 + 45.8i)T^{2} \)
53 \( 1 + (-0.944 + 4.13i)T + (-47.7 - 22.9i)T^{2} \)
59 \( 1 + 11.1T + 59T^{2} \)
61 \( 1 + (4.38 - 2.11i)T + (38.0 - 47.6i)T^{2} \)
67 \( 1 + (-4.45 + 5.58i)T + (-14.9 - 65.3i)T^{2} \)
71 \( 1 + (3.76 + 4.72i)T + (-15.7 + 69.2i)T^{2} \)
73 \( 1 + (2.73 + 11.9i)T + (-65.7 + 31.6i)T^{2} \)
79 \( 1 + (5.86 - 7.35i)T + (-17.5 - 77.0i)T^{2} \)
83 \( 1 + (-11.4 - 5.51i)T + (51.7 + 64.8i)T^{2} \)
89 \( 1 + (-0.398 + 1.74i)T + (-80.1 - 38.6i)T^{2} \)
97 \( 1 + (3.42 + 1.64i)T + (60.4 + 75.8i)T^{2} \)
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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

−10.71587912225879834247094726678, −9.435550884786242042487610187577, −8.952882992718297403858396311560, −7.69111753651128390067772431282, −6.56650057541530579645271122287, −5.62044833438314532497024401419, −4.51123322372680052612577950311, −3.14736503818121022309888496059, −2.43174931719043622804303630478, −0.11469088845825167069727013470, 2.37233655266475913954876742828, 4.00735830828999201012627153299, 4.65004033058177301051120052051, 6.07938331748716522764141869595, 6.81246586935018057291091831924, 7.53207468492190732353879145972, 8.773794919880636393677592204241, 9.467859598669375575608209325789, 10.28537491651074004640096240601, 11.40807592521721833379575859043

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