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Properties

Label 2-68-68.23-c1-0-2
Degree $2$
Conductor $68$
Sign $0.747 + 0.664i$
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  + (−1.30 − 0.541i)2-s + (1.41 + 1.41i)4-s + (1.52 − 2.27i)5-s + (−1.08 − 2.61i)8-s + (2.77 − 1.14i)9-s + (−3.22 + 2.15i)10-s + (−4.23 + 4.23i)13-s + 4i·16-s + (−2.12 + 3.53i)17-s − 4.24·18-s + (5.37 − 1.06i)20-s + (−0.963 − 2.32i)25-s + (7.82 − 3.24i)26-s + (−7.38 − 4.93i)29-s + (2.16 − 5.22i)32-s + ⋯
L(s)  = 1  + (−0.923 − 0.382i)2-s + (0.707 + 0.707i)4-s + (0.681 − 1.01i)5-s + (−0.382 − 0.923i)8-s + (0.923 − 0.382i)9-s + (−1.01 + 0.681i)10-s + (−1.17 + 1.17i)13-s + i·16-s + (−0.514 + 0.857i)17-s − 0.999·18-s + (1.20 − 0.239i)20-s + (−0.192 − 0.465i)25-s + (1.53 − 0.635i)26-s + (−1.37 − 0.916i)29-s + (0.382 − 0.923i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.747 + 0.664i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.747 + 0.664i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(68\)    =    \(2^{2} \cdot 17\)
Sign: $0.747 + 0.664i$
Analytic conductor: \(0.542982\)
Root analytic conductor: \(0.736873\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{68} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 68,\ (\ :1/2),\ 0.747 + 0.664i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.642980 - 0.244466i\)
\(L(\frac12)\) \(\approx\) \(0.642980 - 0.244466i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.30 + 0.541i)T \)
17 \( 1 + (2.12 - 3.53i)T \)
good3 \( 1 + (-2.77 + 1.14i)T^{2} \)
5 \( 1 + (-1.52 + 2.27i)T + (-1.91 - 4.61i)T^{2} \)
7 \( 1 + (2.67 - 6.46i)T^{2} \)
11 \( 1 + (10.1 + 4.20i)T^{2} \)
13 \( 1 + (4.23 - 4.23i)T - 13iT^{2} \)
19 \( 1 + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (-21.2 - 8.80i)T^{2} \)
29 \( 1 + (7.38 + 4.93i)T + (11.0 + 26.7i)T^{2} \)
31 \( 1 + (28.6 - 11.8i)T^{2} \)
37 \( 1 + (-4.90 + 0.976i)T + (34.1 - 14.1i)T^{2} \)
41 \( 1 + (-7.08 - 10.6i)T + (-15.6 + 37.8i)T^{2} \)
43 \( 1 + (-30.4 + 30.4i)T^{2} \)
47 \( 1 + 47iT^{2} \)
53 \( 1 + (13.3 + 5.53i)T + (37.4 + 37.4i)T^{2} \)
59 \( 1 + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (-8.16 + 5.45i)T + (23.3 - 56.3i)T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 + (-65.5 + 27.1i)T^{2} \)
73 \( 1 + (-9.24 + 13.8i)T + (-27.9 - 67.4i)T^{2} \)
79 \( 1 + (72.9 + 30.2i)T^{2} \)
83 \( 1 + (58.6 + 58.6i)T^{2} \)
89 \( 1 + (7.74 + 7.74i)T + 89iT^{2} \)
97 \( 1 + (-3.60 - 2.40i)T + (37.1 + 89.6i)T^{2} \)
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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

−14.82903962345620814948961417422, −13.11497569590252836757329653990, −12.49623307967472077881561954388, −11.24085714959288597913127087214, −9.647580981218837523815643398329, −9.354733040934702358806837654724, −7.82032339059800125710113999234, −6.42713117395466674848683025912, −4.38767234089293368624285196838, −1.81975576926694687013002472494, 2.46186946411692945780557281025, 5.36468325906560154190876237959, 6.87379829195161841642997937943, 7.65177117321070089092788143058, 9.426626163309314978865482399278, 10.23617239270803651014961314138, 11.08477826583535756679761877744, 12.75369247749050366047369420471, 14.16185375476540922372204376296, 15.03993584626555075367671728095

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