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Properties

Label 2-68-17.8-c1-0-0
Degree $2$
Conductor $68$
Sign $0.673 - 0.739i$
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  + (−0.707 + 1.70i)3-s + (1.70 + 0.707i)5-s + (−0.707 + 0.292i)7-s + (−0.292 − 0.292i)9-s + (−1.29 − 3.12i)11-s − 6.82i·13-s + (−2.41 + 2.41i)15-s + (−3 + 2.82i)17-s + (1.82 − 1.82i)19-s − 1.41i·21-s + (2.70 + 6.53i)23-s + (−1.12 − 1.12i)25-s + (−4.41 + 1.82i)27-s + (3.70 + 1.53i)29-s + (2.46 − 5.94i)31-s + ⋯
L(s)  = 1  + (−0.408 + 0.985i)3-s + (0.763 + 0.316i)5-s + (−0.267 + 0.110i)7-s + (−0.0976 − 0.0976i)9-s + (−0.389 − 0.941i)11-s − 1.89i·13-s + (−0.623 + 0.623i)15-s + (−0.727 + 0.685i)17-s + (0.419 − 0.419i)19-s − 0.308i·21-s + (0.564 + 1.36i)23-s + (−0.224 − 0.224i)25-s + (−0.849 + 0.351i)27-s + (0.688 + 0.285i)29-s + (0.442 − 1.06i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.673 - 0.739i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.673 - 0.739i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(68\)    =    \(2^{2} \cdot 17\)
Sign: $0.673 - 0.739i$
Analytic conductor: \(0.542982\)
Root analytic conductor: \(0.736873\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{68} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 68,\ (\ :1/2),\ 0.673 - 0.739i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.797869 + 0.352451i\)
\(L(\frac12)\) \(\approx\) \(0.797869 + 0.352451i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.707 - 1.70i)T + (-2.12 - 2.12i)T^{2} \)
5 \( 1 + (-1.70 - 0.707i)T + (3.53 + 3.53i)T^{2} \)
7 \( 1 + (0.707 - 0.292i)T + (4.94 - 4.94i)T^{2} \)
11 \( 1 + (1.29 + 3.12i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 + 6.82iT - 13T^{2} \)
19 \( 1 + (-1.82 + 1.82i)T - 19iT^{2} \)
23 \( 1 + (-2.70 - 6.53i)T + (-16.2 + 16.2i)T^{2} \)
29 \( 1 + (-3.70 - 1.53i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 + (-2.46 + 5.94i)T + (-21.9 - 21.9i)T^{2} \)
37 \( 1 + (2.53 - 6.12i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (1.12 - 0.464i)T + (28.9 - 28.9i)T^{2} \)
43 \( 1 + (3 + 3i)T + 43iT^{2} \)
47 \( 1 - 3.65iT - 47T^{2} \)
53 \( 1 + (4.17 - 4.17i)T - 53iT^{2} \)
59 \( 1 + (9.82 + 9.82i)T + 59iT^{2} \)
61 \( 1 + (-1.70 + 0.707i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 - 11.3T + 67T^{2} \)
71 \( 1 + (-0.464 + 1.12i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (7.94 + 3.29i)T + (51.6 + 51.6i)T^{2} \)
79 \( 1 + (-4.36 - 10.5i)T + (-55.8 + 55.8i)T^{2} \)
83 \( 1 + (-0.171 + 0.171i)T - 83iT^{2} \)
89 \( 1 - 10.8iT - 89T^{2} \)
97 \( 1 + (-0.878 - 0.363i)T + (68.5 + 68.5i)T^{2} \)
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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

−15.40132835653962360460594876545, −13.73958032815447927284473328196, −12.95425752248519693889156067604, −11.20074640964964706197888432543, −10.42466352951954903108543070191, −9.572820911866977685923009640212, −8.033588893932366431015613392400, −6.11193752601653147817088402257, −5.13766716852691811893987991834, −3.18677286085018356830008348986, 1.92581570181030534121082534710, 4.72246403555887154284439105108, 6.43462212308324563771444126981, 7.14835787081882586610265405171, 8.964899599624275300457487343367, 10.01616532766683053465096737738, 11.61760764063259763592545531530, 12.51428318807862588065609877361, 13.42325583864829813305200235042, 14.32601928163858196478046495378

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