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Properties

Label 2-69-69.68-c1-0-0
Degree $2$
Conductor $69$
Sign $-0.609 - 0.792i$
Analytic cond. $0.550967$
Root an. cond. $0.742272$
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  + 1.84i·2-s + (−1.37 + 1.05i)3-s − 1.39·4-s + (−1.94 − 2.53i)6-s + 1.10i·8-s + (0.771 − 2.89i)9-s + (1.92 − 1.47i)12-s + 7.03·13-s − 4.84·16-s + (5.34 + 1.42i)18-s − 4.79i·23-s + (−1.17 − 1.52i)24-s − 5·25-s + 12.9i·26-s + (1.99 + 4.79i)27-s + ⋯
L(s)  = 1  + 1.30i·2-s + (−0.792 + 0.609i)3-s − 0.699·4-s + (−0.794 − 1.03i)6-s + 0.392i·8-s + (0.257 − 0.966i)9-s + (0.554 − 0.426i)12-s + 1.95·13-s − 1.21·16-s + (1.25 + 0.335i)18-s − 0.999i·23-s + (−0.238 − 0.310i)24-s − 25-s + 2.54i·26-s + (0.384 + 0.922i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.609 - 0.792i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.609 - 0.792i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(69\)    =    \(3 \cdot 23\)
Sign: $-0.609 - 0.792i$
Analytic conductor: \(0.550967\)
Root analytic conductor: \(0.742272\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{69} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 69,\ (\ :1/2),\ -0.609 - 0.792i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.343141 + 0.696524i\)
\(L(\frac12)\) \(\approx\) \(0.343141 + 0.696524i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.37 - 1.05i)T \)
23 \( 1 + 4.79iT \)
good2 \( 1 - 1.84iT - 2T^{2} \)
5 \( 1 + 5T^{2} \)
7 \( 1 - 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 7.03T + 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 19T^{2} \)
29 \( 1 + 10.0iT - 29T^{2} \)
31 \( 1 + 5.83T + 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 - 2.64iT - 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 - 13.7iT - 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + 9.59iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 - 1.04iT - 71T^{2} \)
73 \( 1 + 9.44T + 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−15.58057949603517637849189978537, −14.41056503204311549349828684533, −13.22759171937811686860977380615, −11.62978020740216549536799268924, −10.74745147598566878911865476710, −9.221398329483759492533479485699, −8.010252325936754420631075237797, −6.43377571319064060198559125906, −5.75815579350626966047076266268, −4.17779343243913727442041511497, 1.55299249300073336255592343752, 3.70606195478842663425752487009, 5.69670628217488235399376793487, 7.10634518351940479789103619501, 8.827966650527510871999251728631, 10.39316574009550267253897774995, 11.15041724393293941495677874212, 11.95113835174676491837646187790, 13.04984504002027599821257609800, 13.70011691420372641134363041669

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