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Properties

Label 2-93-31.25-c1-0-2
Degree $2$
Conductor $93$
Sign $0.970 - 0.242i$
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  + 1.41·2-s + (0.5 + 0.866i)3-s + (0.292 − 0.507i)5-s + (0.707 + 1.22i)6-s + (−0.414 − 0.717i)7-s − 2.82·8-s + (−0.499 + 0.866i)9-s + (0.414 − 0.717i)10-s + (0.707 − 1.22i)11-s + (−0.914 + 1.58i)13-s + (−0.585 − 1.01i)14-s + 0.585·15-s − 4.00·16-s + (−3.41 − 5.91i)17-s + (−0.707 + 1.22i)18-s + (1.91 + 3.31i)19-s + ⋯
L(s)  = 1  + 1.00·2-s + (0.288 + 0.499i)3-s + (0.130 − 0.226i)5-s + (0.288 + 0.499i)6-s + (−0.156 − 0.271i)7-s − 0.999·8-s + (−0.166 + 0.288i)9-s + (0.130 − 0.226i)10-s + (0.213 − 0.369i)11-s + (−0.253 + 0.439i)13-s + (−0.156 − 0.271i)14-s + 0.151·15-s − 1.00·16-s + (−0.828 − 1.43i)17-s + (−0.166 + 0.288i)18-s + (0.439 + 0.760i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.242i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 - 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(93\)    =    \(3 \cdot 31\)
Sign: $0.970 - 0.242i$
Analytic conductor: \(0.742608\)
Root analytic conductor: \(0.861747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{93} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 93,\ (\ :1/2),\ 0.970 - 0.242i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.46559 + 0.180399i\)
\(L(\frac12)\) \(\approx\) \(1.46559 + 0.180399i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 + (-2 - 5.19i)T \)
good2 \( 1 - 1.41T + 2T^{2} \)
5 \( 1 + (-0.292 + 0.507i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (0.414 + 0.717i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.707 + 1.22i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.914 - 1.58i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (3.41 + 5.91i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.91 - 3.31i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 3.65T + 23T^{2} \)
29 \( 1 + 3.41T + 29T^{2} \)
37 \( 1 + (0.5 + 0.866i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.12 + 3.67i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.74 - 4.75i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 0.585T + 47T^{2} \)
53 \( 1 + (-0.707 + 1.22i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.24 - 7.34i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + 12.4T + 61T^{2} \)
67 \( 1 + (-7.24 + 12.5i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (3 - 5.19i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (1.91 - 3.31i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (7.82 + 13.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.70 + 6.42i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 17.8T + 89T^{2} \)
97 \( 1 + 11.4T + 97T^{2} \)
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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

−13.97652924015122908283175114083, −13.33496981375261418883763952896, −12.16166490830753545131643454150, −11.08424635813987597392345968480, −9.551148655278432142021552503611, −8.838338733983678075484158834137, −7.04512507900628950389740040420, −5.48811830444857701244763166876, −4.44414736524114581409135922584, −3.11659399706562214283306977111, 2.70264775805852543913746433403, 4.27182809646997239431232546136, 5.76099326410773989471592702905, 6.86586141385464181620628753788, 8.443315895157869941596059591180, 9.526950384266782277444079270360, 11.07487457170894034921082232364, 12.37339448980200700589641765536, 12.98257558744411947757943108786, 13.87047444895435793464353396077

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