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Properties

Label 2-93-93.89-c1-0-6
Degree $2$
Conductor $93$
Sign $0.693 + 0.720i$
Analytic cond. $0.742608$
Root an. cond. $0.861747$
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  + (1.99 − 0.649i)2-s + (−1.51 − 0.836i)3-s + (1.95 − 1.42i)4-s + 0.445i·5-s + (−3.57 − 0.687i)6-s + (1.36 − 0.994i)7-s + (0.515 − 0.710i)8-s + (1.59 + 2.53i)9-s + (0.289 + 0.889i)10-s + (−4.29 + 3.11i)11-s + (−4.15 + 0.518i)12-s + (−0.520 − 0.169i)13-s + (2.08 − 2.87i)14-s + (0.372 − 0.674i)15-s + (−0.924 + 2.84i)16-s + (1.54 + 1.12i)17-s + ⋯
L(s)  = 1  + (1.41 − 0.459i)2-s + (−0.875 − 0.483i)3-s + (0.977 − 0.710i)4-s + 0.199i·5-s + (−1.45 − 0.280i)6-s + (0.517 − 0.375i)7-s + (0.182 − 0.251i)8-s + (0.533 + 0.846i)9-s + (0.0914 + 0.281i)10-s + (−1.29 + 0.940i)11-s + (−1.19 + 0.149i)12-s + (−0.144 − 0.0469i)13-s + (0.558 − 0.768i)14-s + (0.0961 − 0.174i)15-s + (−0.231 + 0.711i)16-s + (0.374 + 0.272i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.693 + 0.720i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.693 + 0.720i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(93\)    =    \(3 \cdot 31\)
Sign: $0.693 + 0.720i$
Analytic conductor: \(0.742608\)
Root analytic conductor: \(0.861747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{93} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 93,\ (\ :1/2),\ 0.693 + 0.720i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.37702 - 0.586169i\)
\(L(\frac12)\) \(\approx\) \(1.37702 - 0.586169i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.51 + 0.836i)T \)
31 \( 1 + (-1.92 + 5.22i)T \)
good2 \( 1 + (-1.99 + 0.649i)T + (1.61 - 1.17i)T^{2} \)
5 \( 1 - 0.445iT - 5T^{2} \)
7 \( 1 + (-1.36 + 0.994i)T + (2.16 - 6.65i)T^{2} \)
11 \( 1 + (4.29 - 3.11i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (0.520 + 0.169i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (-1.54 - 1.12i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.89 + 5.81i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-1.07 - 0.784i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (2.68 + 8.27i)T + (-23.4 + 17.0i)T^{2} \)
37 \( 1 - 2.92iT - 37T^{2} \)
41 \( 1 + (4.60 - 1.49i)T + (33.1 - 24.0i)T^{2} \)
43 \( 1 + (3.10 - 1.00i)T + (34.7 - 25.2i)T^{2} \)
47 \( 1 + (9.40 + 3.05i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (-8.51 - 6.18i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-11.4 - 3.71i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + 1.73iT - 61T^{2} \)
67 \( 1 - 2.55T + 67T^{2} \)
71 \( 1 + (-5.94 + 8.18i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (-2.15 - 2.96i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (3.62 - 4.99i)T + (-24.4 - 75.1i)T^{2} \)
83 \( 1 + (1.64 + 5.06i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (-9.55 + 6.94i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (1.70 - 1.23i)T + (29.9 - 92.2i)T^{2} \)
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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

−13.35412849148875229571869828926, −13.08801067781685050406073541519, −11.91309589889514252085506802622, −11.11373841359999400022113244137, −10.19492079531507635301545207986, −7.902275313152216992188645203098, −6.68140864317473680119575702356, −5.28996496203886199018518950912, −4.50846333535086191067711543114, −2.39278063549997470983916913388, 3.43914691725132919592098558477, 5.06227823727676940864779132536, 5.47786028164965306041259016337, 6.82866181613218856496051004727, 8.458209617360584900328695287314, 10.19546481981660985968713015117, 11.26662911734688458864272474705, 12.33564663885771385617135033540, 13.01964464950708944746204941582, 14.34584706535601140917608883921

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