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Properties

Label 2-68-68.11-c1-0-1
Degree $2$
Conductor $68$
Sign $0.431 - 0.902i$
Analytic cond. $0.542982$
Root an. cond. $0.736873$
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  + (0.114 + 1.40i)2-s + (0.935 − 0.186i)3-s + (−1.97 + 0.323i)4-s + (0.763 + 0.510i)5-s + (0.369 + 1.29i)6-s + (−0.225 − 0.337i)7-s + (−0.681 − 2.74i)8-s + (−1.93 + 0.799i)9-s + (−0.631 + 1.13i)10-s + (0.892 − 4.48i)11-s + (−1.78 + 0.669i)12-s + (−1.21 + 1.21i)13-s + (0.450 − 0.357i)14-s + (0.809 + 0.335i)15-s + (3.79 − 1.27i)16-s + (2.80 − 3.02i)17-s + ⋯
L(s)  = 1  + (0.0810 + 0.996i)2-s + (0.540 − 0.107i)3-s + (−0.986 + 0.161i)4-s + (0.341 + 0.228i)5-s + (0.150 + 0.529i)6-s + (−0.0853 − 0.127i)7-s + (−0.240 − 0.970i)8-s + (−0.643 + 0.266i)9-s + (−0.199 + 0.358i)10-s + (0.269 − 1.35i)11-s + (−0.515 + 0.193i)12-s + (−0.337 + 0.337i)13-s + (0.120 − 0.0954i)14-s + (0.208 + 0.0865i)15-s + (0.947 − 0.318i)16-s + (0.679 − 0.733i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.431 - 0.902i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.431 - 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(68\)    =    \(2^{2} \cdot 17\)
Sign: $0.431 - 0.902i$
Analytic conductor: \(0.542982\)
Root analytic conductor: \(0.736873\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{68} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 68,\ (\ :1/2),\ 0.431 - 0.902i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.842099 + 0.530953i\)
\(L(\frac12)\) \(\approx\) \(0.842099 + 0.530953i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.114 - 1.40i)T \)
17 \( 1 + (-2.80 + 3.02i)T \)
good3 \( 1 + (-0.935 + 0.186i)T + (2.77 - 1.14i)T^{2} \)
5 \( 1 + (-0.763 - 0.510i)T + (1.91 + 4.61i)T^{2} \)
7 \( 1 + (0.225 + 0.337i)T + (-2.67 + 6.46i)T^{2} \)
11 \( 1 + (-0.892 + 4.48i)T + (-10.1 - 4.20i)T^{2} \)
13 \( 1 + (1.21 - 1.21i)T - 13iT^{2} \)
19 \( 1 + (1.29 - 3.11i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (1.22 + 0.243i)T + (21.2 + 8.80i)T^{2} \)
29 \( 1 + (3.14 - 4.71i)T + (-11.0 - 26.7i)T^{2} \)
31 \( 1 + (-1.67 - 8.41i)T + (-28.6 + 11.8i)T^{2} \)
37 \( 1 + (-1.30 - 6.56i)T + (-34.1 + 14.1i)T^{2} \)
41 \( 1 + (-1.11 + 0.743i)T + (15.6 - 37.8i)T^{2} \)
43 \( 1 + (-2.03 - 4.91i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 + (9.03 + 9.03i)T + 47iT^{2} \)
53 \( 1 + (-2.36 - 0.977i)T + (37.4 + 37.4i)T^{2} \)
59 \( 1 + (-11.1 + 4.62i)T + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (-2.41 - 3.61i)T + (-23.3 + 56.3i)T^{2} \)
67 \( 1 + 8.38T + 67T^{2} \)
71 \( 1 + (-13.3 + 2.65i)T + (65.5 - 27.1i)T^{2} \)
73 \( 1 + (5.16 + 3.45i)T + (27.9 + 67.4i)T^{2} \)
79 \( 1 + (-1.81 + 9.10i)T + (-72.9 - 30.2i)T^{2} \)
83 \( 1 + (3.70 + 1.53i)T + (58.6 + 58.6i)T^{2} \)
89 \( 1 + (-6.88 - 6.88i)T + 89iT^{2} \)
97 \( 1 + (-4.91 + 7.35i)T + (-37.1 - 89.6i)T^{2} \)
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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

−14.63521620429304743167236680940, −14.15460711901045428324983280189, −13.34780028083415400966926256214, −11.82025267760648611552139687417, −10.18010304019018919053999407624, −8.878942990780654076118401090948, −8.022726996497575049051345859315, −6.58328113844902444762539848536, −5.36231649901853045393200142944, −3.35594995194368132027147786507, 2.33104359564218407443710408860, 4.05640710785226018877560889382, 5.68789419177026368808061643579, 7.87365722099354254931564070283, 9.252507985724368687989510387889, 9.863441971899067346844110000688, 11.33586445183725800432899397970, 12.44336808633352561116496379391, 13.30061503347065566246961331827, 14.54445929196532535768486467316

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