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Properties

Label 2-2e6-8.5-c1-0-0
Degree $2$
Conductor $64$
Sign $0.707 - 0.707i$
Analytic cond. $0.511042$
Root an. cond. $0.714872$
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  + 2i·3-s − 9-s − 6i·11-s − 6·17-s + 2i·19-s + 5·25-s + 4i·27-s + 12·33-s − 6·41-s + 10i·43-s − 7·49-s − 12i·51-s − 4·57-s − 6i·59-s − 14i·67-s + ⋯
L(s)  = 1  + 1.15i·3-s − 0.333·9-s − 1.80i·11-s − 1.45·17-s + 0.458i·19-s + 25-s + 0.769i·27-s + 2.08·33-s − 0.937·41-s + 1.52i·43-s − 49-s − 1.68i·51-s − 0.529·57-s − 0.781i·59-s − 1.71i·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(64\)    =    \(2^{6}\)
Sign: $0.707 - 0.707i$
Analytic conductor: \(0.511042\)
Root analytic conductor: \(0.714872\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{64} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 64,\ (\ :1/2),\ 0.707 - 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.822784 + 0.340808i\)
\(L(\frac12)\) \(\approx\) \(0.822784 + 0.340808i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
good3 \( 1 - 2iT - 3T^{2} \)
5 \( 1 - 5T^{2} \)
7 \( 1 + 7T^{2} \)
11 \( 1 + 6iT - 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + 6T + 17T^{2} \)
19 \( 1 - 2iT - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 - 10iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 6iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + 14iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 2T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 18iT - 83T^{2} \)
89 \( 1 - 18T + 89T^{2} \)
97 \( 1 - 10T + 97T^{2} \)
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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

−15.24061031761495894594432737469, −14.11744779141590763691021382241, −13.01138251672671175392876917229, −11.32208652688223242730440004648, −10.65483692056500405743349122518, −9.334876857942576191292204419070, −8.351563371273259363456532004580, −6.37961207777060964519609837006, −4.85982387481815205334569253252, −3.40294200831321822698668003934, 2.08774265100019936984633203220, 4.66052578287437967441098070770, 6.66435165558541062030334206319, 7.34996069932697346126672149293, 8.852476829446497820961864119358, 10.26762558111546507012807907342, 11.74893177195257215025275776857, 12.71928374289623567725367707515, 13.40416631353195959623048426493, 14.77666960937088281402227755553

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