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Properties

Label 2-68-17.4-c1-0-0
Degree $2$
Conductor $68$
Sign $-0.155 - 0.987i$
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  + (−2.30 + 2.30i)3-s + (−1 + i)5-s + (2.30 + 2.30i)7-s − 7.60i·9-s + (−0.302 − 0.302i)11-s + 2.60·13-s − 4.60i·15-s + (3.60 + 2i)17-s + 0.605i·19-s − 10.6·21-s + (−4.30 − 4.30i)23-s + 3i·25-s + (10.6 + 10.6i)27-s + (−1.60 + 1.60i)29-s + (4.30 − 4.30i)31-s + ⋯
L(s)  = 1  + (−1.32 + 1.32i)3-s + (−0.447 + 0.447i)5-s + (0.870 + 0.870i)7-s − 2.53i·9-s + (−0.0912 − 0.0912i)11-s + 0.722·13-s − 1.18i·15-s + (0.874 + 0.485i)17-s + 0.138i·19-s − 2.31·21-s + (−0.897 − 0.897i)23-s + 0.600i·25-s + (2.04 + 2.04i)27-s + (−0.298 + 0.298i)29-s + (0.772 − 0.772i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.155 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.155 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(68\)    =    \(2^{2} \cdot 17\)
Sign: $-0.155 - 0.987i$
Analytic conductor: \(0.542982\)
Root analytic conductor: \(0.736873\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{68} (21, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 68,\ (\ :1/2),\ -0.155 - 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.407894 + 0.477285i\)
\(L(\frac12)\) \(\approx\) \(0.407894 + 0.477285i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
17 \( 1 + (-3.60 - 2i)T \)
good3 \( 1 + (2.30 - 2.30i)T - 3iT^{2} \)
5 \( 1 + (1 - i)T - 5iT^{2} \)
7 \( 1 + (-2.30 - 2.30i)T + 7iT^{2} \)
11 \( 1 + (0.302 + 0.302i)T + 11iT^{2} \)
13 \( 1 - 2.60T + 13T^{2} \)
19 \( 1 - 0.605iT - 19T^{2} \)
23 \( 1 + (4.30 + 4.30i)T + 23iT^{2} \)
29 \( 1 + (1.60 - 1.60i)T - 29iT^{2} \)
31 \( 1 + (-4.30 + 4.30i)T - 31iT^{2} \)
37 \( 1 + (-3 + 3i)T - 37iT^{2} \)
41 \( 1 + (-1 - i)T + 41iT^{2} \)
43 \( 1 + 3.39iT - 43T^{2} \)
47 \( 1 + 4T + 47T^{2} \)
53 \( 1 + 5.21iT - 53T^{2} \)
59 \( 1 + 8.60iT - 59T^{2} \)
61 \( 1 + (6.21 + 6.21i)T + 61iT^{2} \)
67 \( 1 - 9.21T + 67T^{2} \)
71 \( 1 + (-2.90 + 2.90i)T - 71iT^{2} \)
73 \( 1 + (7 - 7i)T - 73iT^{2} \)
79 \( 1 + (0.302 + 0.302i)T + 79iT^{2} \)
83 \( 1 - 17.8iT - 83T^{2} \)
89 \( 1 - 7.81T + 89T^{2} \)
97 \( 1 + (7.60 - 7.60i)T - 97iT^{2} \)
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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

−15.25178564214217449030842648284, −14.51994436954582005356967456290, −12.39339190777898116057590494789, −11.48052296856076681123478788049, −10.84854756927367720974309736345, −9.704572559390207903783061132760, −8.258963695258059987428193308822, −6.22168522589032787873681967941, −5.21664992830653049200794516599, −3.82875146730062537429303759852, 1.19862930008452944131412119900, 4.65756739636415016297560336170, 5.99065548520853833551027661526, 7.38965817267865948733237564534, 8.104185246790392304945231290280, 10.40516412963074951293580267267, 11.48358401506787684935844334539, 12.08087506560804325566380033773, 13.27819060806288565341054938954, 14.07735752774627432300909271643

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