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Properties

Label 2-164-164.11-c1-0-15
Degree $2$
Conductor $164$
Sign $0.921 + 0.388i$
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  + (0.0433 + 1.41i)2-s + (0.739 − 1.78i)3-s + (−1.99 + 0.122i)4-s + (1.08 − 2.12i)5-s + (2.55 + 0.968i)6-s + (−0.938 − 3.90i)7-s + (−0.259 − 2.81i)8-s + (−0.521 − 0.521i)9-s + (3.05 + 1.43i)10-s + (−2.35 + 2.75i)11-s + (−1.25 + 3.65i)12-s + (2.54 + 4.15i)13-s + (5.48 − 1.49i)14-s + (−2.99 − 3.50i)15-s + (3.96 − 0.489i)16-s + (0.0520 + 0.661i)17-s + ⋯
L(s)  = 1  + (0.0306 + 0.999i)2-s + (0.427 − 1.03i)3-s + (−0.998 + 0.0613i)4-s + (0.484 − 0.951i)5-s + (1.04 + 0.395i)6-s + (−0.354 − 1.47i)7-s + (−0.0918 − 0.995i)8-s + (−0.173 − 0.173i)9-s + (0.965 + 0.455i)10-s + (−0.709 + 0.830i)11-s + (−0.363 + 1.05i)12-s + (0.706 + 1.15i)13-s + (1.46 − 0.399i)14-s + (−0.773 − 0.905i)15-s + (0.992 − 0.122i)16-s + (0.0126 + 0.160i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.921 + 0.388i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.921 + 0.388i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(164\)    =    \(2^{2} \cdot 41\)
Sign: $0.921 + 0.388i$
Analytic conductor: \(1.30954\)
Root analytic conductor: \(1.14435\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{164} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 164,\ (\ :1/2),\ 0.921 + 0.388i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.20524 - 0.244008i\)
\(L(\frac12)\) \(\approx\) \(1.20524 - 0.244008i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0433 - 1.41i)T \)
41 \( 1 + (5.78 - 2.73i)T \)
good3 \( 1 + (-0.739 + 1.78i)T + (-2.12 - 2.12i)T^{2} \)
5 \( 1 + (-1.08 + 2.12i)T + (-2.93 - 4.04i)T^{2} \)
7 \( 1 + (0.938 + 3.90i)T + (-6.23 + 3.17i)T^{2} \)
11 \( 1 + (2.35 - 2.75i)T + (-1.72 - 10.8i)T^{2} \)
13 \( 1 + (-2.54 - 4.15i)T + (-5.90 + 11.5i)T^{2} \)
17 \( 1 + (-0.0520 - 0.661i)T + (-16.7 + 2.65i)T^{2} \)
19 \( 1 + (-0.534 - 0.327i)T + (8.62 + 16.9i)T^{2} \)
23 \( 1 + (-3.84 - 2.79i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (0.0823 - 1.04i)T + (-28.6 - 4.53i)T^{2} \)
31 \( 1 + (1.06 - 3.26i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (3.25 + 10.0i)T + (-29.9 + 21.7i)T^{2} \)
43 \( 1 + (1.20 + 7.57i)T + (-40.8 + 13.2i)T^{2} \)
47 \( 1 + (0.561 - 2.33i)T + (-41.8 - 21.3i)T^{2} \)
53 \( 1 + (-10.8 - 0.850i)T + (52.3 + 8.29i)T^{2} \)
59 \( 1 + (3.02 - 4.15i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (-1.97 + 12.4i)T + (-58.0 - 18.8i)T^{2} \)
67 \( 1 + (9.59 - 8.19i)T + (10.4 - 66.1i)T^{2} \)
71 \( 1 + (-7.99 - 6.83i)T + (11.1 + 70.1i)T^{2} \)
73 \( 1 + (7.30 - 7.30i)T - 73iT^{2} \)
79 \( 1 + (-1.92 - 0.798i)T + (55.8 + 55.8i)T^{2} \)
83 \( 1 - 9.71iT - 83T^{2} \)
89 \( 1 + (4.79 - 1.15i)T + (79.2 - 40.4i)T^{2} \)
97 \( 1 + (-1.98 + 1.69i)T + (15.1 - 95.8i)T^{2} \)
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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

−13.21311817227413166490406145571, −12.41505066076950819022662441392, −10.51524390374011942093834824232, −9.398799488827960520994722635896, −8.447644271681466063213079160444, −7.26536933005779161564926488461, −6.87504996374937179598039576068, −5.25720587859508653613095393967, −4.02629310362175797672059564563, −1.36990364522275914110633743331, 2.77066156746790825761311082636, 3.25172327737745651203397466882, 5.09551091154358474082741286021, 6.10899338057460550526411096121, 8.351178844303846488102644102856, 9.061535328790396062778155425373, 10.14790674356325030039419188169, 10.62388789737048425705822944849, 11.72040119081140303365819503502, 12.92331850944518871361573932690

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