Properties

Label 2-46-23.4-c3-0-0
Degree $2$
Conductor $46$
Sign $-0.963 + 0.266i$
Analytic cond. $2.71408$
Root an. cond. $1.64744$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.830 + 1.81i)2-s + (−5.88 + 1.72i)3-s + (−2.61 + 3.02i)4-s + (−11.4 − 7.33i)5-s + (−8.03 − 9.27i)6-s + (−0.0110 − 0.0771i)7-s + (−7.67 − 2.25i)8-s + (8.95 − 5.75i)9-s + (3.85 − 26.8i)10-s + (−18.6 + 40.8i)11-s + (10.1 − 22.3i)12-s + (−3.95 + 27.5i)13-s + (0.131 − 0.0842i)14-s + (79.8 + 23.4i)15-s + (−2.27 − 15.8i)16-s + (−11.8 − 13.6i)17-s + ⋯
L(s)  = 1  + (0.293 + 0.643i)2-s + (−1.13 + 0.332i)3-s + (−0.327 + 0.377i)4-s + (−1.02 − 0.655i)5-s + (−0.546 − 0.631i)6-s + (−0.000598 − 0.00416i)7-s + (−0.339 − 0.0996i)8-s + (0.331 − 0.213i)9-s + (0.122 − 0.848i)10-s + (−0.510 + 1.11i)11-s + (0.245 − 0.537i)12-s + (−0.0844 + 0.587i)13-s + (0.00250 − 0.00160i)14-s + (1.37 + 0.403i)15-s + (−0.0355 − 0.247i)16-s + (−0.169 − 0.195i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.963 + 0.266i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.963 + 0.266i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(46\)    =    \(2 \cdot 23\)
Sign: $-0.963 + 0.266i$
Analytic conductor: \(2.71408\)
Root analytic conductor: \(1.64744\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{46} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 46,\ (\ :3/2),\ -0.963 + 0.266i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0423408 - 0.311505i\)
\(L(\frac12)\) \(\approx\) \(0.0423408 - 0.311505i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.830 - 1.81i)T \)
23 \( 1 + (-16.7 - 109. i)T \)
good3 \( 1 + (5.88 - 1.72i)T + (22.7 - 14.5i)T^{2} \)
5 \( 1 + (11.4 + 7.33i)T + (51.9 + 113. i)T^{2} \)
7 \( 1 + (0.0110 + 0.0771i)T + (-329. + 96.6i)T^{2} \)
11 \( 1 + (18.6 - 40.8i)T + (-871. - 1.00e3i)T^{2} \)
13 \( 1 + (3.95 - 27.5i)T + (-2.10e3 - 618. i)T^{2} \)
17 \( 1 + (11.8 + 13.6i)T + (-699. + 4.86e3i)T^{2} \)
19 \( 1 + (0.965 - 1.11i)T + (-976. - 6.78e3i)T^{2} \)
29 \( 1 + (178. + 205. i)T + (-3.47e3 + 2.41e4i)T^{2} \)
31 \( 1 + (-294. - 86.5i)T + (2.50e4 + 1.61e4i)T^{2} \)
37 \( 1 + (319. - 205. i)T + (2.10e4 - 4.60e4i)T^{2} \)
41 \( 1 + (251. + 161. i)T + (2.86e4 + 6.26e4i)T^{2} \)
43 \( 1 + (191. - 56.2i)T + (6.68e4 - 4.29e4i)T^{2} \)
47 \( 1 + 380.T + 1.03e5T^{2} \)
53 \( 1 + (-78.7 - 547. i)T + (-1.42e5 + 4.19e4i)T^{2} \)
59 \( 1 + (-17.6 + 122. i)T + (-1.97e5 - 5.78e4i)T^{2} \)
61 \( 1 + (-375. - 110. i)T + (1.90e5 + 1.22e5i)T^{2} \)
67 \( 1 + (-237. - 520. i)T + (-1.96e5 + 2.27e5i)T^{2} \)
71 \( 1 + (-255. - 558. i)T + (-2.34e5 + 2.70e5i)T^{2} \)
73 \( 1 + (-158. + 182. i)T + (-5.53e4 - 3.85e5i)T^{2} \)
79 \( 1 + (-59.1 + 411. i)T + (-4.73e5 - 1.38e5i)T^{2} \)
83 \( 1 + (949. - 609. i)T + (2.37e5 - 5.20e5i)T^{2} \)
89 \( 1 + (121. - 35.6i)T + (5.93e5 - 3.81e5i)T^{2} \)
97 \( 1 + (-280. - 180. i)T + (3.79e5 + 8.30e5i)T^{2} \)
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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

−15.80119056768691766173276800577, −15.30929079131707049458664767522, −13.55344127870060999447454391800, −12.17292636075632613031303232225, −11.59449071943598851069830320515, −9.951225357655008018396142321079, −8.271757549667378074803742408855, −6.94292126041363673443238159751, −5.27159965751773065279910045365, −4.31533550126288989933263240591, 0.25196409826475903033555665905, 3.33786341467040659460211240038, 5.28427206114706819509967698644, 6.69526716888009948836978514234, 8.340811855404222498481617135703, 10.51618081524926063715477960432, 11.19168829486435188875229411714, 12.05301383115094436288975617600, 13.16061834833006269253416223804, 14.63716867916189783529921586463

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