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Properties

Label 2-93-93.23-c1-0-1
Degree $2$
Conductor $93$
Sign $0.275 - 0.961i$
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  + (−0.543 − 0.176i)2-s + (0.0442 + 1.73i)3-s + (−1.35 − 0.983i)4-s + 2.69i·5-s + (0.281 − 0.948i)6-s + (2.73 + 1.98i)7-s + (1.23 + 1.69i)8-s + (−2.99 + 0.153i)9-s + (0.475 − 1.46i)10-s + (−2.18 − 1.58i)11-s + (1.64 − 2.38i)12-s + (1.46 − 0.477i)13-s + (−1.13 − 1.56i)14-s + (−4.66 + 0.119i)15-s + (0.663 + 2.04i)16-s + (3.29 − 2.39i)17-s + ⋯
L(s)  = 1  + (−0.384 − 0.124i)2-s + (0.0255 + 0.999i)3-s + (−0.676 − 0.491i)4-s + 1.20i·5-s + (0.115 − 0.387i)6-s + (1.03 + 0.751i)7-s + (0.436 + 0.600i)8-s + (−0.998 + 0.0510i)9-s + (0.150 − 0.463i)10-s + (−0.657 − 0.477i)11-s + (0.474 − 0.689i)12-s + (0.407 − 0.132i)13-s + (−0.303 − 0.418i)14-s + (−1.20 + 0.0307i)15-s + (0.165 + 0.510i)16-s + (0.800 − 0.581i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.275 - 0.961i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.275 - 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(93\)    =    \(3 \cdot 31\)
Sign: $0.275 - 0.961i$
Analytic conductor: \(0.742608\)
Root analytic conductor: \(0.861747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{93} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 93,\ (\ :1/2),\ 0.275 - 0.961i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.606376 + 0.456873i\)
\(L(\frac12)\) \(\approx\) \(0.606376 + 0.456873i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.0442 - 1.73i)T \)
31 \( 1 + (1.39 - 5.39i)T \)
good2 \( 1 + (0.543 + 0.176i)T + (1.61 + 1.17i)T^{2} \)
5 \( 1 - 2.69iT - 5T^{2} \)
7 \( 1 + (-2.73 - 1.98i)T + (2.16 + 6.65i)T^{2} \)
11 \( 1 + (2.18 + 1.58i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (-1.46 + 0.477i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (-3.29 + 2.39i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-1.90 + 5.87i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (2.52 - 1.83i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (-1.44 + 4.45i)T + (-23.4 - 17.0i)T^{2} \)
37 \( 1 - 4.55iT - 37T^{2} \)
41 \( 1 + (-9.14 - 2.97i)T + (33.1 + 24.0i)T^{2} \)
43 \( 1 + (-2.36 - 0.767i)T + (34.7 + 25.2i)T^{2} \)
47 \( 1 + (3.95 - 1.28i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (-7.14 + 5.18i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (9.92 - 3.22i)T + (47.7 - 34.6i)T^{2} \)
61 \( 1 + 9.83iT - 61T^{2} \)
67 \( 1 + 13.6T + 67T^{2} \)
71 \( 1 + (-2.58 - 3.56i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (3.97 - 5.46i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (0.105 + 0.145i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (-5.28 + 16.2i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (10.7 + 7.77i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-3.32 - 2.41i)T + (29.9 + 92.2i)T^{2} \)
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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

−14.40990871803727728492974623398, −13.67035965849075718802310728417, −11.54395932008779659986123532091, −10.92166646954510354084733255918, −10.00913515399946474050959621230, −8.932557024311151860878567310492, −7.88395569421456440986652147027, −5.80528802642567218323396868978, −4.80980171124277166970807915694, −2.90544884477516597935672811998, 1.27321933192185168637178615572, 4.16402119856500860001503964309, 5.53075650690416204654554546048, 7.63339037973026349120788196296, 7.995032156647786768589018677890, 9.055490258972371417734485560573, 10.53752622423755495830176343800, 12.15010192816331868293148294574, 12.69993498800237357017526467647, 13.68226525818098880827029602520

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