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Properties

Label 2-68-68.3-c1-0-3
Degree $2$
Conductor $68$
Sign $0.751 - 0.660i$
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  + (1.15 + 0.817i)2-s + (−0.0741 + 0.372i)3-s + (0.664 + 1.88i)4-s + (−1.26 − 1.89i)5-s + (−0.390 + 0.369i)6-s + (−3.23 − 2.15i)7-s + (−0.773 + 2.72i)8-s + (2.63 + 1.09i)9-s + (0.0865 − 3.21i)10-s + (1.93 − 0.385i)11-s + (−0.752 + 0.107i)12-s + (−1.26 − 1.26i)13-s + (−1.96 − 5.13i)14-s + (0.798 − 0.330i)15-s + (−3.11 + 2.50i)16-s + (−2.29 + 3.42i)17-s + ⋯
L(s)  = 1  + (0.816 + 0.577i)2-s + (−0.0427 + 0.215i)3-s + (0.332 + 0.943i)4-s + (−0.565 − 0.846i)5-s + (−0.159 + 0.150i)6-s + (−1.22 − 0.815i)7-s + (−0.273 + 0.961i)8-s + (0.879 + 0.364i)9-s + (0.0273 − 1.01i)10-s + (0.584 − 0.116i)11-s + (−0.217 + 0.0311i)12-s + (−0.349 − 0.349i)13-s + (−0.525 − 1.37i)14-s + (0.206 − 0.0854i)15-s + (−0.778 + 0.627i)16-s + (−0.555 + 0.831i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.751 - 0.660i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.751 - 0.660i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(68\)    =    \(2^{2} \cdot 17\)
Sign: $0.751 - 0.660i$
Analytic conductor: \(0.542982\)
Root analytic conductor: \(0.736873\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{68} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 68,\ (\ :1/2),\ 0.751 - 0.660i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.10851 + 0.417903i\)
\(L(\frac12)\) \(\approx\) \(1.10851 + 0.417903i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.15 - 0.817i)T \)
17 \( 1 + (2.29 - 3.42i)T \)
good3 \( 1 + (0.0741 - 0.372i)T + (-2.77 - 1.14i)T^{2} \)
5 \( 1 + (1.26 + 1.89i)T + (-1.91 + 4.61i)T^{2} \)
7 \( 1 + (3.23 + 2.15i)T + (2.67 + 6.46i)T^{2} \)
11 \( 1 + (-1.93 + 0.385i)T + (10.1 - 4.20i)T^{2} \)
13 \( 1 + (1.26 + 1.26i)T + 13iT^{2} \)
19 \( 1 + (1.34 + 3.25i)T + (-13.4 + 13.4i)T^{2} \)
23 \( 1 + (-1.09 - 5.49i)T + (-21.2 + 8.80i)T^{2} \)
29 \( 1 + (2.30 - 1.53i)T + (11.0 - 26.7i)T^{2} \)
31 \( 1 + (-8.35 - 1.66i)T + (28.6 + 11.8i)T^{2} \)
37 \( 1 + (0.599 + 0.119i)T + (34.1 + 14.1i)T^{2} \)
41 \( 1 + (1.13 - 1.69i)T + (-15.6 - 37.8i)T^{2} \)
43 \( 1 + (-3.47 + 8.38i)T + (-30.4 - 30.4i)T^{2} \)
47 \( 1 + (3.89 - 3.89i)T - 47iT^{2} \)
53 \( 1 + (-10.3 + 4.30i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (-1.67 - 0.694i)T + (41.7 + 41.7i)T^{2} \)
61 \( 1 + (7.37 + 4.92i)T + (23.3 + 56.3i)T^{2} \)
67 \( 1 + 13.7T + 67T^{2} \)
71 \( 1 + (-1.76 + 8.86i)T + (-65.5 - 27.1i)T^{2} \)
73 \( 1 + (3.00 + 4.49i)T + (-27.9 + 67.4i)T^{2} \)
79 \( 1 + (-0.434 + 0.0865i)T + (72.9 - 30.2i)T^{2} \)
83 \( 1 + (-11.0 + 4.55i)T + (58.6 - 58.6i)T^{2} \)
89 \( 1 + (0.569 - 0.569i)T - 89iT^{2} \)
97 \( 1 + (-13.4 + 8.97i)T + (37.1 - 89.6i)T^{2} \)
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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

−15.18580900943335695231528581982, −13.53226824457879825435715014951, −13.00358283107846776219884013871, −12.00620729777210975198567971059, −10.48574044194106042197534821446, −9.012645613556096007924450179958, −7.57265769017314074989389079158, −6.49689134834204185833899179150, −4.72199123829042588966711620307, −3.69817599640136736943272171504, 2.76115011268631043260877912388, 4.20390842075521094924851811928, 6.26797378621727791556500229229, 6.99668783594134958346636461472, 9.308424947722797388702459788559, 10.28299671232331047256171262081, 11.69910953916334015818171592942, 12.36398585233521349132097559374, 13.38579101495192770562481959378, 14.72721248809484877879785568234

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