Properties

Label 2-230-23.18-c3-0-19
Degree $2$
Conductor $230$
Sign $0.924 + 0.380i$
Analytic cond. $13.5704$
Root an. cond. $3.68380$
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  + (1.91 + 0.563i)2-s + (6.53 − 7.54i)3-s + (3.36 + 2.16i)4-s + (−0.711 + 4.94i)5-s + (16.7 − 10.7i)6-s + (14.9 + 32.6i)7-s + (5.23 + 6.04i)8-s + (−10.3 − 71.9i)9-s + (−4.15 + 9.09i)10-s + (46.9 − 13.7i)11-s + (38.3 − 11.2i)12-s + (6.38 − 13.9i)13-s + (10.2 + 71.1i)14-s + (32.6 + 37.7i)15-s + (6.64 + 14.5i)16-s + (−68.0 + 43.7i)17-s + ⋯
L(s)  = 1  + (0.678 + 0.199i)2-s + (1.25 − 1.45i)3-s + (0.420 + 0.270i)4-s + (−0.0636 + 0.442i)5-s + (1.14 − 0.734i)6-s + (0.806 + 1.76i)7-s + (0.231 + 0.267i)8-s + (−0.383 − 2.66i)9-s + (−0.131 + 0.287i)10-s + (1.28 − 0.377i)11-s + (0.921 − 0.270i)12-s + (0.136 − 0.298i)13-s + (0.195 + 1.35i)14-s + (0.562 + 0.649i)15-s + (0.103 + 0.227i)16-s + (−0.971 + 0.624i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.924 + 0.380i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.924 + 0.380i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $0.924 + 0.380i$
Analytic conductor: \(13.5704\)
Root analytic conductor: \(3.68380\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :3/2),\ 0.924 + 0.380i)\)

Particular Values

\(L(2)\) \(\approx\) \(4.19251 - 0.828988i\)
\(L(\frac12)\) \(\approx\) \(4.19251 - 0.828988i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.91 - 0.563i)T \)
5 \( 1 + (0.711 - 4.94i)T \)
23 \( 1 + (91.8 - 61.0i)T \)
good3 \( 1 + (-6.53 + 7.54i)T + (-3.84 - 26.7i)T^{2} \)
7 \( 1 + (-14.9 - 32.6i)T + (-224. + 259. i)T^{2} \)
11 \( 1 + (-46.9 + 13.7i)T + (1.11e3 - 719. i)T^{2} \)
13 \( 1 + (-6.38 + 13.9i)T + (-1.43e3 - 1.66e3i)T^{2} \)
17 \( 1 + (68.0 - 43.7i)T + (2.04e3 - 4.46e3i)T^{2} \)
19 \( 1 + (34.4 + 22.1i)T + (2.84e3 + 6.23e3i)T^{2} \)
29 \( 1 + (-129. + 83.2i)T + (1.01e4 - 2.21e4i)T^{2} \)
31 \( 1 + (119. + 138. i)T + (-4.23e3 + 2.94e4i)T^{2} \)
37 \( 1 + (4.51 + 31.4i)T + (-4.86e4 + 1.42e4i)T^{2} \)
41 \( 1 + (-54.5 + 379. i)T + (-6.61e4 - 1.94e4i)T^{2} \)
43 \( 1 + (25.7 - 29.6i)T + (-1.13e4 - 7.86e4i)T^{2} \)
47 \( 1 + 145.T + 1.03e5T^{2} \)
53 \( 1 + (-113. - 248. i)T + (-9.74e4 + 1.12e5i)T^{2} \)
59 \( 1 + (103. - 227. i)T + (-1.34e5 - 1.55e5i)T^{2} \)
61 \( 1 + (189. + 218. i)T + (-3.23e4 + 2.24e5i)T^{2} \)
67 \( 1 + (375. + 110. i)T + (2.53e5 + 1.62e5i)T^{2} \)
71 \( 1 + (-309. - 90.9i)T + (3.01e5 + 1.93e5i)T^{2} \)
73 \( 1 + (-225. - 145. i)T + (1.61e5 + 3.53e5i)T^{2} \)
79 \( 1 + (223. - 489. i)T + (-3.22e5 - 3.72e5i)T^{2} \)
83 \( 1 + (-78.5 - 546. i)T + (-5.48e5 + 1.61e5i)T^{2} \)
89 \( 1 + (-384. + 444. i)T + (-1.00e5 - 6.97e5i)T^{2} \)
97 \( 1 + (53.6 - 373. i)T + (-8.75e5 - 2.57e5i)T^{2} \)
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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

−12.00859852140368916593226769023, −11.32110076428467252127066804972, −9.163479419179471899785906328265, −8.571768459689849381105519516153, −7.75018048795320564675248108528, −6.51734783307630845801118816668, −5.90536434237375878051598860115, −3.87823986072200324362995240761, −2.55716472070541335219561316311, −1.79366395001307808335692689945, 1.73004020071256099798439782807, 3.49828914912966041355982580870, 4.41571325687728944363700745486, 4.65886470507943297697103752810, 6.84482670770083459251116558024, 8.040326021816453527437973194585, 8.993387225677902571076565189249, 9.993081142963050700991893141473, 10.73001325933411091666493601023, 11.58438735752270953154992973962

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