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Properties

Label 2-68-68.63-c1-0-2
Degree $2$
Conductor $68$
Sign $0.386 - 0.922i$
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  + (0.541 + 1.30i)2-s + (−1.41 + 1.41i)4-s + (0.548 + 0.109i)5-s + (−2.61 − 1.08i)8-s + (1.14 − 2.77i)9-s + (0.154 + 0.775i)10-s + (−2.83 − 2.83i)13-s − 4i·16-s + (2.12 + 3.53i)17-s + 4.24·18-s + (−0.930 + 0.621i)20-s + (−4.33 − 1.79i)25-s + (2.17 − 5.24i)26-s + (−2.06 + 10.3i)29-s + (5.22 − 2.16i)32-s + ⋯
L(s)  = 1  + (0.382 + 0.923i)2-s + (−0.707 + 0.707i)4-s + (0.245 + 0.0488i)5-s + (−0.923 − 0.382i)8-s + (0.382 − 0.923i)9-s + (0.0488 + 0.245i)10-s + (−0.786 − 0.786i)13-s − i·16-s + (0.514 + 0.857i)17-s + 0.999·18-s + (−0.207 + 0.138i)20-s + (−0.866 − 0.358i)25-s + (0.425 − 1.02i)26-s + (−0.383 + 1.92i)29-s + (0.923 − 0.382i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(68\)    =    \(2^{2} \cdot 17\)
Sign: $0.386 - 0.922i$
Analytic conductor: \(0.542982\)
Root analytic conductor: \(0.736873\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{68} (63, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 68,\ (\ :1/2),\ 0.386 - 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.854935 + 0.568482i\)
\(L(\frac12)\) \(\approx\) \(0.854935 + 0.568482i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.541 - 1.30i)T \)
17 \( 1 + (-2.12 - 3.53i)T \)
good3 \( 1 + (-1.14 + 2.77i)T^{2} \)
5 \( 1 + (-0.548 - 0.109i)T + (4.61 + 1.91i)T^{2} \)
7 \( 1 + (-6.46 + 2.67i)T^{2} \)
11 \( 1 + (4.20 + 10.1i)T^{2} \)
13 \( 1 + (2.83 + 2.83i)T + 13iT^{2} \)
19 \( 1 + (13.4 - 13.4i)T^{2} \)
23 \( 1 + (-8.80 - 21.2i)T^{2} \)
29 \( 1 + (2.06 - 10.3i)T + (-26.7 - 11.0i)T^{2} \)
31 \( 1 + (11.8 - 28.6i)T^{2} \)
37 \( 1 + (9.46 - 6.32i)T + (14.1 - 34.1i)T^{2} \)
41 \( 1 + (-9.60 + 1.91i)T + (37.8 - 15.6i)T^{2} \)
43 \( 1 + (30.4 + 30.4i)T^{2} \)
47 \( 1 - 47iT^{2} \)
53 \( 1 + (0.636 + 1.53i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (-41.7 - 41.7i)T^{2} \)
61 \( 1 + (3.03 + 15.2i)T + (-56.3 + 23.3i)T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 + (-27.1 + 65.5i)T^{2} \)
73 \( 1 + (-8.83 - 1.75i)T + (67.4 + 27.9i)T^{2} \)
79 \( 1 + (30.2 + 72.9i)T^{2} \)
83 \( 1 + (-58.6 + 58.6i)T^{2} \)
89 \( 1 + (-10.8 + 10.8i)T - 89iT^{2} \)
97 \( 1 + (3.24 - 16.3i)T + (-89.6 - 37.1i)T^{2} \)
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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

−14.98723473331147489056124906960, −14.17086521963669027959146848929, −12.82468694051780509916609009343, −12.18536906106280271415036705947, −10.26458552423912401471523968535, −9.094428032323487135227972027546, −7.75750867597590078313370943881, −6.54411990407674774833668925684, −5.27595257444523631367961490649, −3.57668046060518159524349880959, 2.25908052345647285250453586604, 4.30560896781543036109008417982, 5.60543590610537199809302526035, 7.52136702132043242405694583276, 9.256917733822909580424169533259, 10.13481883588743389732572353874, 11.35092395951444067325127770854, 12.31427619893651557624076807408, 13.53651661713325298708659670509, 14.15148346115691602551281767385

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