↑

Properties

Label 2-68-17.2-c1-0-0
Degree $2$
Conductor $68$
Sign $0.997 + 0.0758i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 − 0.292i)3-s + (0.292 + 0.707i)5-s + (0.707 − 1.70i)7-s + (−1.70 + 1.70i)9-s + (−2.70 − 1.12i)11-s + 1.17i·13-s + (0.414 + 0.414i)15-s + (−3 + 2.82i)17-s + (−3.82 − 3.82i)19-s − 1.41i·21-s + (1.29 + 0.535i)23-s + (3.12 − 3.12i)25-s + (−1.58 + 3.82i)27-s + (2.29 + 5.53i)29-s + (9.53 − 3.94i)31-s + ⋯
L(s)  = 1  + (0.408 − 0.169i)3-s + (0.130 + 0.316i)5-s + (0.267 − 0.645i)7-s + (−0.569 + 0.569i)9-s + (−0.816 − 0.338i)11-s + 0.324i·13-s + (0.106 + 0.106i)15-s + (−0.727 + 0.685i)17-s + (−0.878 − 0.878i)19-s − 0.308i·21-s + (0.269 + 0.111i)23-s + (0.624 − 0.624i)25-s + (−0.305 + 0.736i)27-s + (0.425 + 1.02i)29-s + (1.71 − 0.709i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0758i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0758i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(68\)    =    \(2^{2} \cdot 17\)
Sign: $0.997 + 0.0758i$
Analytic conductor: \(0.542982\)
Root analytic conductor: \(0.736873\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{68} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 68,\ (\ :1/2),\ 0.997 + 0.0758i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.983679 - 0.0373776i\)
\(L(\frac12)\) \(\approx\) \(0.983679 - 0.0373776i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3 - 2.82i)T \)
good3 \( 1 + (-0.707 + 0.292i)T + (2.12 - 2.12i)T^{2} \)
5 \( 1 + (-0.292 - 0.707i)T + (-3.53 + 3.53i)T^{2} \)
7 \( 1 + (-0.707 + 1.70i)T + (-4.94 - 4.94i)T^{2} \)
11 \( 1 + (2.70 + 1.12i)T + (7.77 + 7.77i)T^{2} \)
13 \( 1 - 1.17iT - 13T^{2} \)
19 \( 1 + (3.82 + 3.82i)T + 19iT^{2} \)
23 \( 1 + (-1.29 - 0.535i)T + (16.2 + 16.2i)T^{2} \)
29 \( 1 + (-2.29 - 5.53i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 + (-9.53 + 3.94i)T + (21.9 - 21.9i)T^{2} \)
37 \( 1 + (-4.53 + 1.87i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (-3.12 + 7.53i)T + (-28.9 - 28.9i)T^{2} \)
43 \( 1 + (3 - 3i)T - 43iT^{2} \)
47 \( 1 - 7.65iT - 47T^{2} \)
53 \( 1 + (9.82 + 9.82i)T + 53iT^{2} \)
59 \( 1 + (4.17 - 4.17i)T - 59iT^{2} \)
61 \( 1 + (-0.292 + 0.707i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 + 11.3T + 67T^{2} \)
71 \( 1 + (-7.53 + 3.12i)T + (50.2 - 50.2i)T^{2} \)
73 \( 1 + (-1.94 - 4.70i)T + (-51.6 + 51.6i)T^{2} \)
79 \( 1 + (8.36 + 3.46i)T + (55.8 + 55.8i)T^{2} \)
83 \( 1 + (-5.82 - 5.82i)T + 83iT^{2} \)
89 \( 1 + 5.17iT - 89T^{2} \)
97 \( 1 + (-5.12 - 12.3i)T + (-68.5 + 68.5i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.59248980693893062182556363897, −13.72881120618198097170641282737, −12.87808970222774331392770546407, −11.14856766551788593128448670233, −10.51908972774685416669574999444, −8.822403613378010779262915473224, −7.82475344117225822797783741755, −6.42787307705634405116482479614, −4.64829835418288073902502923827, −2.62823533973929041948162804329, 2.72972076859317181103470803703, 4.77012835759556907836848933910, 6.22824171043440741531865820494, 8.053233581116513543384450012809, 8.948928087728075650892011618770, 10.16070168558060873378136844634, 11.57608224669363984929993002516, 12.62404238671691107149644519504, 13.75073489872135485390499677632, 14.97672703310955132387119154884

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