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Properties

Label 2-93-93.92-c1-0-0
Degree $2$
Conductor $93$
Sign $-0.953 - 0.300i$
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  + 2.07i·2-s + (−1.46 + 0.921i)3-s − 2.30·4-s + 0.628i·5-s + (−1.91 − 3.04i)6-s − 7-s − 0.628i·8-s + (1.30 − 2.70i)9-s − 1.30·10-s + 2.04·11-s + (3.37 − 2.12i)12-s + 4.24i·13-s − 2.07i·14-s + (−0.578 − 0.921i)15-s − 3.30·16-s + 4.70·17-s + ⋯
L(s)  = 1  + 1.46i·2-s + (−0.846 + 0.531i)3-s − 1.15·4-s + 0.280i·5-s + (−0.780 − 1.24i)6-s − 0.377·7-s − 0.222i·8-s + (0.434 − 0.900i)9-s − 0.411·10-s + 0.616·11-s + (0.975 − 0.612i)12-s + 1.17i·13-s − 0.554i·14-s + (−0.149 − 0.237i)15-s − 0.825·16-s + 1.14·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.953 - 0.300i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.953 - 0.300i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(93\)    =    \(3 \cdot 31\)
Sign: $-0.953 - 0.300i$
Analytic conductor: \(0.742608\)
Root analytic conductor: \(0.861747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{93} (92, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 93,\ (\ :1/2),\ -0.953 - 0.300i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.114500 + 0.743494i\)
\(L(\frac12)\) \(\approx\) \(0.114500 + 0.743494i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.46 - 0.921i)T \)
31 \( 1 + (3.60 + 4.24i)T \)
good2 \( 1 - 2.07iT - 2T^{2} \)
5 \( 1 - 0.628iT - 5T^{2} \)
7 \( 1 + T + 7T^{2} \)
11 \( 1 - 2.04T + 11T^{2} \)
13 \( 1 - 4.24iT - 13T^{2} \)
17 \( 1 - 4.70T + 17T^{2} \)
19 \( 1 - 3.60T + 19T^{2} \)
23 \( 1 + 6.75T + 23T^{2} \)
29 \( 1 - 6.75T + 29T^{2} \)
37 \( 1 + 9.76iT - 37T^{2} \)
41 \( 1 + 6.03iT - 41T^{2} \)
43 \( 1 + 4.24iT - 43T^{2} \)
47 \( 1 - 5.40iT - 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 - 0.628iT - 59T^{2} \)
61 \( 1 - 5.52iT - 61T^{2} \)
67 \( 1 - 6.60T + 67T^{2} \)
71 \( 1 - 10.1iT - 71T^{2} \)
73 \( 1 - 9.76iT - 73T^{2} \)
79 \( 1 + 14.0iT - 79T^{2} \)
83 \( 1 + 4.09T + 83T^{2} \)
89 \( 1 + 4.70T + 89T^{2} \)
97 \( 1 - 12.8T + 97T^{2} \)
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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

−14.56680196631699691985025179431, −14.08700153749467855355340575445, −12.33589657847918311815574782558, −11.38482714285353653858627339304, −9.961774257226974689523928369614, −8.974178728351688739135292117465, −7.37176417137233614528108669872, −6.44416457210393762785668280345, −5.51252521902475065628374961435, −4.07315297549381631426820202726, 1.16518409788272296386321400514, 3.22018347103901639595994041873, 4.99873902202124013892787910560, 6.48172459718623728522947864131, 8.069197412908330961111267318847, 9.775237412476073449535940313203, 10.44603140000526207236159817010, 11.66900981938131403611658303092, 12.28953052547002256716499133120, 13.01471844036772902422198053466

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