Properties

Label 2-68-68.27-c1-0-3
Degree $2$
Conductor $68$
Sign $0.999 + 0.0261i$
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  + (−1.08 + 0.910i)2-s + (0.861 − 1.28i)3-s + (0.343 − 1.97i)4-s + (1.07 − 0.213i)5-s + (0.240 + 2.17i)6-s + (0.127 − 0.641i)7-s + (1.42 + 2.44i)8-s + (0.228 + 0.552i)9-s + (−0.967 + 1.20i)10-s + (−3.00 + 2.01i)11-s + (−2.24 − 2.13i)12-s + (1.76 − 1.76i)13-s + (0.445 + 0.809i)14-s + (0.649 − 1.56i)15-s + (−3.76 − 1.35i)16-s + (−4.12 − 0.0241i)17-s + ⋯
L(s)  = 1  + (−0.765 + 0.643i)2-s + (0.497 − 0.744i)3-s + (0.171 − 0.985i)4-s + (0.480 − 0.0955i)5-s + (0.0983 + 0.889i)6-s + (0.0481 − 0.242i)7-s + (0.502 + 0.864i)8-s + (0.0762 + 0.184i)9-s + (−0.306 + 0.382i)10-s + (−0.907 + 0.606i)11-s + (−0.647 − 0.617i)12-s + (0.488 − 0.488i)13-s + (0.119 + 0.216i)14-s + (0.167 − 0.404i)15-s + (−0.941 − 0.338i)16-s + (−0.999 − 0.00585i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0261i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0261i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(68\)    =    \(2^{2} \cdot 17\)
Sign: $0.999 + 0.0261i$
Analytic conductor: \(0.542982\)
Root analytic conductor: \(0.736873\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{68} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 68,\ (\ :1/2),\ 0.999 + 0.0261i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.774287 - 0.0101204i\)
\(L(\frac12)\) \(\approx\) \(0.774287 - 0.0101204i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.08 - 0.910i)T \)
17 \( 1 + (4.12 + 0.0241i)T \)
good3 \( 1 + (-0.861 + 1.28i)T + (-1.14 - 2.77i)T^{2} \)
5 \( 1 + (-1.07 + 0.213i)T + (4.61 - 1.91i)T^{2} \)
7 \( 1 + (-0.127 + 0.641i)T + (-6.46 - 2.67i)T^{2} \)
11 \( 1 + (3.00 - 2.01i)T + (4.20 - 10.1i)T^{2} \)
13 \( 1 + (-1.76 + 1.76i)T - 13iT^{2} \)
19 \( 1 + (5.03 + 2.08i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (-5.17 - 7.75i)T + (-8.80 + 21.2i)T^{2} \)
29 \( 1 + (0.213 + 1.07i)T + (-26.7 + 11.0i)T^{2} \)
31 \( 1 + (-0.159 - 0.106i)T + (11.8 + 28.6i)T^{2} \)
37 \( 1 + (-3.44 - 2.29i)T + (14.1 + 34.1i)T^{2} \)
41 \( 1 + (8.33 + 1.65i)T + (37.8 + 15.6i)T^{2} \)
43 \( 1 + (-3.49 + 1.44i)T + (30.4 - 30.4i)T^{2} \)
47 \( 1 + (6.02 + 6.02i)T + 47iT^{2} \)
53 \( 1 + (-4.17 + 10.0i)T + (-37.4 - 37.4i)T^{2} \)
59 \( 1 + (0.832 + 2.00i)T + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (-1.06 + 5.37i)T + (-56.3 - 23.3i)T^{2} \)
67 \( 1 - 6.44T + 67T^{2} \)
71 \( 1 + (2.02 - 3.02i)T + (-27.1 - 65.5i)T^{2} \)
73 \( 1 + (0.183 - 0.0364i)T + (67.4 - 27.9i)T^{2} \)
79 \( 1 + (-1.38 + 0.926i)T + (30.2 - 72.9i)T^{2} \)
83 \( 1 + (-4.62 + 11.1i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (3.79 + 3.79i)T + 89iT^{2} \)
97 \( 1 + (-3.00 - 15.1i)T + (-89.6 + 37.1i)T^{2} \)
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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

−15.01722640176333734248275118312, −13.46422686501704516572820739620, −13.20652509040793048873515796242, −11.12636238326849956037583854697, −10.06916995073173331854933524706, −8.767873382537452874977725459530, −7.73729371757478104583675924469, −6.74375311003277069715723290406, −5.16383863309987229900702517374, −2.03919032930974263781579135419, 2.60480072413881245021048445243, 4.25237807371468819665627261041, 6.50076571748384032037722986786, 8.385188015475125446288813314211, 9.069065434699849684672306966603, 10.28645133111847104747486692492, 11.01706506303096964000547311853, 12.53956617771181310464235406406, 13.56144796197425764253355973737, 14.95932499750837891663389005716

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