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Properties

Label 2-230-5.4-c5-0-41
Degree $2$
Conductor $230$
Sign $0.249 + 0.968i$
Analytic cond. $36.8882$
Root an. cond. $6.07357$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4i·2-s − 24.8i·3-s − 16·4-s + (13.9 + 54.1i)5-s + 99.4·6-s − 51.4i·7-s − 64i·8-s − 375.·9-s + (−216. + 55.6i)10-s + 150.·11-s + 397. i·12-s + 560. i·13-s + 205.·14-s + (1.34e3 − 346. i)15-s + 256·16-s − 1.84e3i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s − 1.59i·3-s − 0.5·4-s + (0.249 + 0.968i)5-s + 1.12·6-s − 0.396i·7-s − 0.353i·8-s − 1.54·9-s + (−0.684 + 0.176i)10-s + 0.374·11-s + 0.797i·12-s + 0.920i·13-s + 0.280·14-s + (1.54 − 0.397i)15-s + 0.250·16-s − 1.54i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.249 + 0.968i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.249 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $0.249 + 0.968i$
Analytic conductor: \(36.8882\)
Root analytic conductor: \(6.07357\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :5/2),\ 0.249 + 0.968i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.689661131\)
\(L(\frac12)\) \(\approx\) \(1.689661131\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 4iT \)
5 \( 1 + (-13.9 - 54.1i)T \)
23 \( 1 - 529iT \)
good3 \( 1 + 24.8iT - 243T^{2} \)
7 \( 1 + 51.4iT - 1.68e4T^{2} \)
11 \( 1 - 150.T + 1.61e5T^{2} \)
13 \( 1 - 560. iT - 3.71e5T^{2} \)
17 \( 1 + 1.84e3iT - 1.41e6T^{2} \)
19 \( 1 - 2.48e3T + 2.47e6T^{2} \)
29 \( 1 - 841.T + 2.05e7T^{2} \)
31 \( 1 - 1.65e3T + 2.86e7T^{2} \)
37 \( 1 + 6.02e3iT - 6.93e7T^{2} \)
41 \( 1 + 2.13e4T + 1.15e8T^{2} \)
43 \( 1 + 1.26e4iT - 1.47e8T^{2} \)
47 \( 1 + 1.54e4iT - 2.29e8T^{2} \)
53 \( 1 + 2.41e4iT - 4.18e8T^{2} \)
59 \( 1 - 1.47e4T + 7.14e8T^{2} \)
61 \( 1 + 1.65e4T + 8.44e8T^{2} \)
67 \( 1 + 4.32e4iT - 1.35e9T^{2} \)
71 \( 1 - 5.09e4T + 1.80e9T^{2} \)
73 \( 1 + 6.42e4iT - 2.07e9T^{2} \)
79 \( 1 + 1.05e5T + 3.07e9T^{2} \)
83 \( 1 + 3.89e4iT - 3.93e9T^{2} \)
89 \( 1 - 1.04e5T + 5.58e9T^{2} \)
97 \( 1 - 5.28e4iT - 8.58e9T^{2} \)
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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

−11.55461730124162598100719258160, −10.02013919106063260951245700979, −8.964814077477054666761857356633, −7.58949340447985426997848583716, −7.09125279883892708067915382576, −6.49756904037609135815275921441, −5.25624486886675879475432889935, −3.36309131470587559734814942519, −1.94458435065209526690819907301, −0.55265697543803756043453480570, 1.22315219748750338931326860673, 3.03997163176610646395167582902, 4.08553979750998310080545655204, 5.05590873425362246365176992840, 5.83969835782190423946708089514, 8.173413205600650491426386574743, 8.900975912744268835776017594149, 9.816286037619917109466312001876, 10.31596572469336124537110208628, 11.44467814444891990383788475386

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