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Properties

Label 2-164-164.79-c1-0-1
Degree $2$
Conductor $164$
Sign $-0.742 - 0.670i$
Analytic cond. $1.30954$
Root an. cond. $1.14435$
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  + (−1.38 − 0.300i)2-s + (−1.22 + 2.95i)3-s + (1.81 + 0.830i)4-s + (1.90 + 1.90i)5-s + (2.58 − 3.72i)6-s + (−0.394 + 0.951i)7-s + (−2.26 − 1.69i)8-s + (−5.13 − 5.13i)9-s + (−2.05 − 3.19i)10-s + (−0.00775 − 0.00321i)11-s + (−4.68 + 4.36i)12-s + (4.09 + 1.69i)13-s + (0.830 − 1.19i)14-s + (−7.96 + 3.29i)15-s + (2.62 + 3.02i)16-s + (−6.47 + 2.68i)17-s + ⋯
L(s)  = 1  + (−0.977 − 0.212i)2-s + (−0.707 + 1.70i)3-s + (0.909 + 0.415i)4-s + (0.850 + 0.850i)5-s + (1.05 − 1.51i)6-s + (−0.148 + 0.359i)7-s + (−0.800 − 0.599i)8-s + (−1.71 − 1.71i)9-s + (−0.650 − 1.01i)10-s + (−0.00233 − 0.000968i)11-s + (−1.35 + 1.26i)12-s + (1.13 + 0.470i)13-s + (0.221 − 0.319i)14-s + (−2.05 + 0.851i)15-s + (0.655 + 0.755i)16-s + (−1.57 + 0.650i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.742 - 0.670i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.742 - 0.670i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(164\)    =    \(2^{2} \cdot 41\)
Sign: $-0.742 - 0.670i$
Analytic conductor: \(1.30954\)
Root analytic conductor: \(1.14435\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{164} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 164,\ (\ :1/2),\ -0.742 - 0.670i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.221834 + 0.576666i\)
\(L(\frac12)\) \(\approx\) \(0.221834 + 0.576666i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.38 + 0.300i)T \)
41 \( 1 + (-4.81 - 4.21i)T \)
good3 \( 1 + (1.22 - 2.95i)T + (-2.12 - 2.12i)T^{2} \)
5 \( 1 + (-1.90 - 1.90i)T + 5iT^{2} \)
7 \( 1 + (0.394 - 0.951i)T + (-4.94 - 4.94i)T^{2} \)
11 \( 1 + (0.00775 + 0.00321i)T + (7.77 + 7.77i)T^{2} \)
13 \( 1 + (-4.09 - 1.69i)T + (9.19 + 9.19i)T^{2} \)
17 \( 1 + (6.47 - 2.68i)T + (12.0 - 12.0i)T^{2} \)
19 \( 1 + (1.58 + 3.81i)T + (-13.4 + 13.4i)T^{2} \)
23 \( 1 - 3.15T + 23T^{2} \)
29 \( 1 + (3.54 + 1.46i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 - 5.01T + 31T^{2} \)
37 \( 1 + 0.251T + 37T^{2} \)
43 \( 1 + (-6.40 - 6.40i)T + 43iT^{2} \)
47 \( 1 + (1.11 + 2.70i)T + (-33.2 + 33.2i)T^{2} \)
53 \( 1 + (1.42 - 3.43i)T + (-37.4 - 37.4i)T^{2} \)
59 \( 1 - 1.28iT - 59T^{2} \)
61 \( 1 + (-4.53 + 4.53i)T - 61iT^{2} \)
67 \( 1 + (-4.06 - 9.82i)T + (-47.3 + 47.3i)T^{2} \)
71 \( 1 + (3.01 - 7.28i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (-6.56 + 6.56i)T - 73iT^{2} \)
79 \( 1 + (11.8 + 4.90i)T + (55.8 + 55.8i)T^{2} \)
83 \( 1 + 5.13iT - 83T^{2} \)
89 \( 1 + (-2.01 - 0.836i)T + (62.9 + 62.9i)T^{2} \)
97 \( 1 + (2.61 + 6.30i)T + (-68.5 + 68.5i)T^{2} \)
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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

−13.09861394027593489010304924251, −11.41045389802499455255952525734, −11.02217802045807409777781591606, −10.28853050679772031205207386210, −9.315427219265950212446523740339, −8.749502077927871372487718314547, −6.56269222559871124895234893800, −5.99075426313155954862295697992, −4.21492685056719115978646726872, −2.71226701528114447573938175058, 0.884402412970308622227270204803, 2.11262564250282743301765499399, 5.44758688746140542423671628220, 6.27238977509092091502785220461, 7.14118876731712931906146152835, 8.290791898658803675453812711384, 9.099333125217175074464877482658, 10.61694692814308634426138896776, 11.38371104201152195088100560352, 12.54559322467243436104348929911

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