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Properties

Label 2-230-5.4-c1-0-2
Degree $2$
Conductor $230$
Sign $0.818 - 0.574i$
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  − i·2-s + 0.706i·3-s − 4-s + (−1.83 + 1.28i)5-s + 0.706·6-s + 2.40i·7-s + i·8-s + 2.50·9-s + (1.28 + 1.83i)10-s + 4.95·11-s − 0.706i·12-s + 4.40i·13-s + 2.40·14-s + (−0.907 − 1.29i)15-s + 16-s + 0.200i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.407i·3-s − 0.5·4-s + (−0.818 + 0.574i)5-s + 0.288·6-s + 0.910i·7-s + 0.353i·8-s + 0.833·9-s + (0.406 + 0.578i)10-s + 1.49·11-s − 0.203i·12-s + 1.22i·13-s + 0.643·14-s + (−0.234 − 0.333i)15-s + 0.250·16-s + 0.0487i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.818 - 0.574i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.818 - 0.574i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $0.818 - 0.574i$
Analytic conductor: \(1.83655\)
Root analytic conductor: \(1.35519\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :1/2),\ 0.818 - 0.574i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.02197 + 0.322737i\)
\(L(\frac12)\) \(\approx\) \(1.02197 + 0.322737i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
5 \( 1 + (1.83 - 1.28i)T \)
23 \( 1 - iT \)
good3 \( 1 - 0.706iT - 3T^{2} \)
7 \( 1 - 2.40iT - 7T^{2} \)
11 \( 1 - 4.95T + 11T^{2} \)
13 \( 1 - 4.40iT - 13T^{2} \)
17 \( 1 - 0.200iT - 17T^{2} \)
19 \( 1 + 6.65T + 19T^{2} \)
29 \( 1 + 1.67T + 29T^{2} \)
31 \( 1 + 1.82T + 31T^{2} \)
37 \( 1 + 8.47iT - 37T^{2} \)
41 \( 1 - 11.0T + 41T^{2} \)
43 \( 1 + 2.06iT - 43T^{2} \)
47 \( 1 - 0.505iT - 47T^{2} \)
53 \( 1 + 5.84iT - 53T^{2} \)
59 \( 1 - 4.41T + 59T^{2} \)
61 \( 1 + 6.67T + 61T^{2} \)
67 \( 1 - 12.6iT - 67T^{2} \)
71 \( 1 - 3.61T + 71T^{2} \)
73 \( 1 + 14.4iT - 73T^{2} \)
79 \( 1 + 9.95T + 79T^{2} \)
83 \( 1 + 13.3iT - 83T^{2} \)
89 \( 1 - 5.04T + 89T^{2} \)
97 \( 1 - 3.79iT - 97T^{2} \)
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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

−12.08085133675126116914484304791, −11.40333009258722067983446246335, −10.56138019774622063165515970620, −9.316084581584396969773829885843, −8.809262491833417325716319758551, −7.27000787807939045964972856450, −6.21270320324507708544048852796, −4.38235360910346612960507068713, −3.81125538327995495762438563178, −2.05116450071420519732852063230, 1.00572717173770733123118737618, 3.84378001871445421598620003941, 4.56260562566894265018566213416, 6.23071891074332904671067875551, 7.17420809202471850008927472190, 7.944729221120869846975897043960, 8.915408238906878907004802894966, 10.09803092250194828920831716099, 11.18995021458947763488194558192, 12.49310476556318172318526829255

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