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Properties

Label 2-122-61.20-c1-0-0
Degree $2$
Conductor $122$
Sign $-0.396 - 0.917i$
Analytic cond. $0.974174$
Root an. cond. $0.987002$
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 + (0.215 + 0.662i)3-s + (−0.809 + 0.587i)4-s + (−3.11 + 2.26i)5-s + (−0.563 + 0.409i)6-s + (−0.440 + 1.35i)7-s + (−0.809 − 0.587i)8-s + (2.03 − 1.47i)9-s + (−3.11 − 2.26i)10-s + 4.34·11-s + (−0.563 − 0.409i)12-s + 4.28·13-s − 1.42·14-s + (−2.17 − 1.57i)15-s + (0.309 − 0.951i)16-s + (−2.32 + 1.68i)17-s + ⋯
L(s)  = 1  + (0.218 + 0.672i)2-s + (0.124 + 0.382i)3-s + (−0.404 + 0.293i)4-s + (−1.39 + 1.01i)5-s + (−0.230 + 0.167i)6-s + (−0.166 + 0.512i)7-s + (−0.286 − 0.207i)8-s + (0.678 − 0.492i)9-s + (−0.985 − 0.715i)10-s + 1.30·11-s + (−0.162 − 0.118i)12-s + 1.18·13-s − 0.381·14-s + (−0.560 − 0.407i)15-s + (0.0772 − 0.237i)16-s + (−0.563 + 0.409i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 122 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.396 - 0.917i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.396 - 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(122\)    =    \(2 \cdot 61\)
Sign: $-0.396 - 0.917i$
Analytic conductor: \(0.974174\)
Root analytic conductor: \(0.987002\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{122} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 122,\ (\ :1/2),\ -0.396 - 0.917i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.560063 + 0.852277i\)
\(L(\frac12)\) \(\approx\) \(0.560063 + 0.852277i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.309 - 0.951i)T \)
61 \( 1 + (-4.32 + 6.50i)T \)
good3 \( 1 + (-0.215 - 0.662i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (3.11 - 2.26i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (0.440 - 1.35i)T + (-5.66 - 4.11i)T^{2} \)
11 \( 1 - 4.34T + 11T^{2} \)
13 \( 1 - 4.28T + 13T^{2} \)
17 \( 1 + (2.32 - 1.68i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (1.43 + 4.41i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (4.27 - 3.10i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 - 3.96T + 29T^{2} \)
31 \( 1 + (1.49 - 4.61i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-3.39 + 10.4i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-0.435 + 1.34i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + (0.248 + 0.180i)T + (13.2 + 40.8i)T^{2} \)
47 \( 1 + 11.5T + 47T^{2} \)
53 \( 1 + (-3.59 - 2.61i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (3.35 + 10.3i)T + (-47.7 + 34.6i)T^{2} \)
67 \( 1 + (3.06 - 2.22i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (-7.29 - 5.30i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (0.176 + 0.128i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (7.95 + 5.78i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-2.25 - 6.93i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (1.49 + 4.59i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (1.12 - 3.46i)T + (-78.4 - 57.0i)T^{2} \)
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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

−14.15360103386893730800771604621, −12.71846930909315505299326494539, −11.70072515195455458824972209521, −10.83755197433284205642117860373, −9.309830711230348525243835736054, −8.367649055078122589016004424705, −7.01538153752379937857647121172, −6.32984274299999981858203332669, −4.21996655811123517783490694787, −3.52256322750072149974526519631, 1.24671262588085451666271463421, 3.85222848354421730594303388567, 4.46624978257028856957199752446, 6.50683101938713187847351317667, 7.953968740113158321660398749167, 8.721070080712412788486798433527, 10.11445853819009287830752514433, 11.40949725758453115933034666579, 12.05552160817397680922852888345, 13.00539886276288076955998436452

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