Properties

Label 2-46-23.2-c1-0-1
Degree $2$
Conductor $46$
Sign $0.957 - 0.289i$
Analytic cond. $0.367311$
Root an. cond. $0.606062$
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  + (0.841 + 0.540i)2-s + (−0.0530 − 0.368i)3-s + (0.415 + 0.909i)4-s + (−3.26 − 0.957i)5-s + (0.154 − 0.339i)6-s + (−0.297 + 0.342i)7-s + (−0.142 + 0.989i)8-s + (2.74 − 0.806i)9-s + (−2.22 − 2.56i)10-s + (−2.53 + 1.63i)11-s + (0.313 − 0.201i)12-s + (−0.592 − 0.683i)13-s + (−0.435 + 0.127i)14-s + (−0.180 + 1.25i)15-s + (−0.654 + 0.755i)16-s + (2.61 − 5.72i)17-s + ⋯
L(s)  = 1  + (0.594 + 0.382i)2-s + (−0.0306 − 0.213i)3-s + (0.207 + 0.454i)4-s + (−1.45 − 0.428i)5-s + (0.0632 − 0.138i)6-s + (−0.112 + 0.129i)7-s + (−0.0503 + 0.349i)8-s + (0.915 − 0.268i)9-s + (−0.703 − 0.812i)10-s + (−0.765 + 0.491i)11-s + (0.0905 − 0.0581i)12-s + (−0.164 − 0.189i)13-s + (−0.116 + 0.0341i)14-s + (−0.0465 + 0.323i)15-s + (−0.163 + 0.188i)16-s + (0.634 − 1.38i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.289i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.957 - 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(46\)    =    \(2 \cdot 23\)
Sign: $0.957 - 0.289i$
Analytic conductor: \(0.367311\)
Root analytic conductor: \(0.606062\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{46} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 46,\ (\ :1/2),\ 0.957 - 0.289i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.905786 + 0.133741i\)
\(L(\frac12)\) \(\approx\) \(0.905786 + 0.133741i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.841 - 0.540i)T \)
23 \( 1 + (0.838 - 4.72i)T \)
good3 \( 1 + (0.0530 + 0.368i)T + (-2.87 + 0.845i)T^{2} \)
5 \( 1 + (3.26 + 0.957i)T + (4.20 + 2.70i)T^{2} \)
7 \( 1 + (0.297 - 0.342i)T + (-0.996 - 6.92i)T^{2} \)
11 \( 1 + (2.53 - 1.63i)T + (4.56 - 10.0i)T^{2} \)
13 \( 1 + (0.592 + 0.683i)T + (-1.85 + 12.8i)T^{2} \)
17 \( 1 + (-2.61 + 5.72i)T + (-11.1 - 12.8i)T^{2} \)
19 \( 1 + (-2.73 - 5.99i)T + (-12.4 + 14.3i)T^{2} \)
29 \( 1 + (0.645 - 1.41i)T + (-18.9 - 21.9i)T^{2} \)
31 \( 1 + (-1.07 + 7.47i)T + (-29.7 - 8.73i)T^{2} \)
37 \( 1 + (1.19 - 0.349i)T + (31.1 - 20.0i)T^{2} \)
41 \( 1 + (7.37 + 2.16i)T + (34.4 + 22.1i)T^{2} \)
43 \( 1 + (0.887 + 6.16i)T + (-41.2 + 12.1i)T^{2} \)
47 \( 1 + 10.0T + 47T^{2} \)
53 \( 1 + (-6.13 + 7.07i)T + (-7.54 - 52.4i)T^{2} \)
59 \( 1 + (-4.43 - 5.11i)T + (-8.39 + 58.3i)T^{2} \)
61 \( 1 + (0.447 - 3.11i)T + (-58.5 - 17.1i)T^{2} \)
67 \( 1 + (5.96 + 3.83i)T + (27.8 + 60.9i)T^{2} \)
71 \( 1 + (-10.6 - 6.87i)T + (29.4 + 64.5i)T^{2} \)
73 \( 1 + (0.587 + 1.28i)T + (-47.8 + 55.1i)T^{2} \)
79 \( 1 + (-0.618 - 0.713i)T + (-11.2 + 78.1i)T^{2} \)
83 \( 1 + (-7.57 + 2.22i)T + (69.8 - 44.8i)T^{2} \)
89 \( 1 + (-1.58 - 11.0i)T + (-85.3 + 25.0i)T^{2} \)
97 \( 1 + (1.40 + 0.412i)T + (81.6 + 52.4i)T^{2} \)
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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

−15.77694395908843110216859916534, −15.01785233848387058905864995239, −13.47460614551475789709052087214, −12.32118960619796727925145108915, −11.74286012446323817114544049006, −9.844985406150764802304119578597, −7.931339989233421033632675683587, −7.27091024785774862353565890335, −5.18266757085814367251493104799, −3.70978694412176825466196417233, 3.39927698431355657306873936164, 4.75984945987412289484106013577, 6.88290268186084724706764575593, 8.176828644816406683998054792628, 10.22501587517462627112564806622, 11.09998727543153182699193430822, 12.26412074858985745088761044173, 13.32278522294395907181944714538, 14.79637471900099175489028532502, 15.61218657249840073849056567514

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