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Properties

Label 2-230-115.9-c1-0-2
Degree $2$
Conductor $230$
Sign $0.903 - 0.429i$
Analytic cond. $1.83655$
Root an. cond. $1.35519$
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  + (−0.281 − 0.959i)2-s + (−0.0697 + 0.0604i)3-s + (−0.841 + 0.540i)4-s + (−0.948 + 2.02i)5-s + (0.0776 + 0.0499i)6-s + (1.04 + 0.475i)7-s + (0.755 + 0.654i)8-s + (−0.425 + 2.96i)9-s + (2.21 + 0.339i)10-s + (3.69 + 1.08i)11-s + (0.0260 − 0.0886i)12-s + (1.25 − 0.571i)13-s + (0.162 − 1.13i)14-s + (−0.0563 − 0.198i)15-s + (0.415 − 0.909i)16-s + (−0.463 + 0.720i)17-s + ⋯
L(s)  = 1  + (−0.199 − 0.678i)2-s + (−0.0402 + 0.0349i)3-s + (−0.420 + 0.270i)4-s + (−0.424 + 0.905i)5-s + (0.0317 + 0.0203i)6-s + (0.393 + 0.179i)7-s + (0.267 + 0.231i)8-s + (−0.141 + 0.987i)9-s + (0.698 + 0.107i)10-s + (1.11 + 0.327i)11-s + (0.00751 − 0.0255i)12-s + (0.346 − 0.158i)13-s + (0.0435 − 0.302i)14-s + (−0.0145 − 0.0513i)15-s + (0.103 − 0.227i)16-s + (−0.112 + 0.174i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.903 - 0.429i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.903 - 0.429i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $0.903 - 0.429i$
Analytic conductor: \(1.83655\)
Root analytic conductor: \(1.35519\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :1/2),\ 0.903 - 0.429i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.994995 + 0.224395i\)
\(L(\frac12)\) \(\approx\) \(0.994995 + 0.224395i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.281 + 0.959i)T \)
5 \( 1 + (0.948 - 2.02i)T \)
23 \( 1 + (-0.229 - 4.79i)T \)
good3 \( 1 + (0.0697 - 0.0604i)T + (0.426 - 2.96i)T^{2} \)
7 \( 1 + (-1.04 - 0.475i)T + (4.58 + 5.29i)T^{2} \)
11 \( 1 + (-3.69 - 1.08i)T + (9.25 + 5.94i)T^{2} \)
13 \( 1 + (-1.25 + 0.571i)T + (8.51 - 9.82i)T^{2} \)
17 \( 1 + (0.463 - 0.720i)T + (-7.06 - 15.4i)T^{2} \)
19 \( 1 + (0.778 - 0.500i)T + (7.89 - 17.2i)T^{2} \)
29 \( 1 + (-0.00469 - 0.00301i)T + (12.0 + 26.3i)T^{2} \)
31 \( 1 + (-4.36 + 5.04i)T + (-4.41 - 30.6i)T^{2} \)
37 \( 1 + (-1.06 - 0.153i)T + (35.5 + 10.4i)T^{2} \)
41 \( 1 + (0.702 + 4.88i)T + (-39.3 + 11.5i)T^{2} \)
43 \( 1 + (5.24 - 4.54i)T + (6.11 - 42.5i)T^{2} \)
47 \( 1 - 7.27iT - 47T^{2} \)
53 \( 1 + (-0.417 - 0.190i)T + (34.7 + 40.0i)T^{2} \)
59 \( 1 + (-0.895 - 1.95i)T + (-38.6 + 44.5i)T^{2} \)
61 \( 1 + (-6.33 + 7.30i)T + (-8.68 - 60.3i)T^{2} \)
67 \( 1 + (2.18 + 7.42i)T + (-56.3 + 36.2i)T^{2} \)
71 \( 1 + (-3.45 + 1.01i)T + (59.7 - 38.3i)T^{2} \)
73 \( 1 + (7.64 + 11.8i)T + (-30.3 + 66.4i)T^{2} \)
79 \( 1 + (0.879 + 1.92i)T + (-51.7 + 59.7i)T^{2} \)
83 \( 1 + (-10.1 - 1.46i)T + (79.6 + 23.3i)T^{2} \)
89 \( 1 + (4.67 + 5.39i)T + (-12.6 + 88.0i)T^{2} \)
97 \( 1 + (-19.0 + 2.73i)T + (93.0 - 27.3i)T^{2} \)
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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

−11.86682893487158111688546639731, −11.35747198565814517913386645738, −10.53495822450192252427176934012, −9.554514965699543272068051956009, −8.327508529027022009789497949960, −7.48771560633743559198052972301, −6.18071567555820015679087033808, −4.63705547455001636079046910197, −3.44216107694410968462113789677, −1.96691833332836969688956229680, 1.02308278683945208608112464870, 3.76849410432037319178504013477, 4.78911827485337292581013691397, 6.14550904237592394538723947347, 7.02902329967776255446719944532, 8.480736746832486784602889213639, 8.810153696621469804423200561542, 9.944249926613605384986907097592, 11.39329293750408796026296560807, 12.08041567962809512489480503017

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