Properties

Label 2-164-164.15-c1-0-6
Degree $2$
Conductor $164$
Sign $-0.0267 - 0.999i$
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 + 0.270i)2-s + (0.869 + 2.09i)3-s + (1.85 − 0.751i)4-s + (1.95 + 3.84i)5-s + (−1.77 − 2.67i)6-s + (0.861 − 3.58i)7-s + (−2.36 + 1.54i)8-s + (−1.52 + 1.52i)9-s + (−3.75 − 4.80i)10-s + (−1.33 − 1.56i)11-s + (3.18 + 3.23i)12-s + (−0.539 + 0.880i)13-s + (−0.225 + 5.21i)14-s + (−6.35 + 7.44i)15-s + (2.87 − 2.78i)16-s + (0.209 − 2.66i)17-s + ⋯
L(s)  = 1  + (−0.981 + 0.191i)2-s + (0.501 + 1.21i)3-s + (0.926 − 0.375i)4-s + (0.875 + 1.71i)5-s + (−0.724 − 1.09i)6-s + (0.325 − 1.35i)7-s + (−0.837 + 0.545i)8-s + (−0.509 + 0.509i)9-s + (−1.18 − 1.51i)10-s + (−0.403 − 0.472i)11-s + (0.920 + 0.934i)12-s + (−0.149 + 0.244i)13-s + (−0.0601 + 1.39i)14-s + (−1.64 + 1.92i)15-s + (0.717 − 0.696i)16-s + (0.0509 − 0.647i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0267 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0267 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(164\)    =    \(2^{2} \cdot 41\)
Sign: $-0.0267 - 0.999i$
Analytic conductor: \(1.30954\)
Root analytic conductor: \(1.14435\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{164} (15, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 164,\ (\ :1/2),\ -0.0267 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.695691 + 0.714587i\)
\(L(\frac12)\) \(\approx\) \(0.695691 + 0.714587i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.38 - 0.270i)T \)
41 \( 1 + (-6.09 + 1.95i)T \)
good3 \( 1 + (-0.869 - 2.09i)T + (-2.12 + 2.12i)T^{2} \)
5 \( 1 + (-1.95 - 3.84i)T + (-2.93 + 4.04i)T^{2} \)
7 \( 1 + (-0.861 + 3.58i)T + (-6.23 - 3.17i)T^{2} \)
11 \( 1 + (1.33 + 1.56i)T + (-1.72 + 10.8i)T^{2} \)
13 \( 1 + (0.539 - 0.880i)T + (-5.90 - 11.5i)T^{2} \)
17 \( 1 + (-0.209 + 2.66i)T + (-16.7 - 2.65i)T^{2} \)
19 \( 1 + (-0.0153 + 0.00939i)T + (8.62 - 16.9i)T^{2} \)
23 \( 1 + (3.82 - 2.78i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (0.347 + 4.41i)T + (-28.6 + 4.53i)T^{2} \)
31 \( 1 + (2.93 + 9.03i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-1.09 + 3.36i)T + (-29.9 - 21.7i)T^{2} \)
43 \( 1 + (0.739 - 4.66i)T + (-40.8 - 13.2i)T^{2} \)
47 \( 1 + (-0.161 - 0.671i)T + (-41.8 + 21.3i)T^{2} \)
53 \( 1 + (-0.621 + 0.0489i)T + (52.3 - 8.29i)T^{2} \)
59 \( 1 + (1.67 + 2.31i)T + (-18.2 + 56.1i)T^{2} \)
61 \( 1 + (0.0314 + 0.198i)T + (-58.0 + 18.8i)T^{2} \)
67 \( 1 + (-7.84 - 6.69i)T + (10.4 + 66.1i)T^{2} \)
71 \( 1 + (-1.54 + 1.31i)T + (11.1 - 70.1i)T^{2} \)
73 \( 1 + (-3.34 - 3.34i)T + 73iT^{2} \)
79 \( 1 + (1.95 - 0.808i)T + (55.8 - 55.8i)T^{2} \)
83 \( 1 - 14.8iT - 83T^{2} \)
89 \( 1 + (-3.06 - 0.735i)T + (79.2 + 40.4i)T^{2} \)
97 \( 1 + (2.16 + 1.84i)T + (15.1 + 95.8i)T^{2} \)
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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

−13.66350458507695436375206683746, −11.19279721238121843080644627945, −10.87553337134434097809361427859, −9.796657810909821967163262959457, −9.658263669442254186259489901891, −7.87239018544449087378690349539, −7.04049635538343498614519101521, −5.80720161772746564609044811419, −3.82452813826489011027915254907, −2.51814976894510888806584606415, 1.51071304293979501957969435105, 2.34254142311087673456067218046, 5.20322546242777927968230335057, 6.32205994997838355357454703963, 7.84414718189301181945554711806, 8.572108600871949728680335988246, 9.108517930607833784644364246890, 10.30419249337023911872089349476, 12.11321771295241024790733541462, 12.45254832986822154342222344745

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