Properties

Label 2-230-5.4-c5-0-31
Degree $2$
Conductor $230$
Sign $0.500 + 0.865i$
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 − 5.96i·3-s − 16·4-s + (28.0 + 48.3i)5-s − 23.8·6-s − 183. i·7-s + 64i·8-s + 207.·9-s + (193. − 112. i)10-s + 76.8·11-s + 95.3i·12-s + 418. i·13-s − 734.·14-s + (288. − 166. i)15-s + 256·16-s + 1.76e3i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.382i·3-s − 0.5·4-s + (0.500 + 0.865i)5-s − 0.270·6-s − 1.41i·7-s + 0.353i·8-s + 0.853·9-s + (0.612 − 0.354i)10-s + 0.191·11-s + 0.191i·12-s + 0.686i·13-s − 1.00·14-s + (0.331 − 0.191i)15-s + 0.250·16-s + 1.47i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.500 + 0.865i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.500 + 0.865i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $0.500 + 0.865i$
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: \((2,\ 230,\ (\ :5/2),\ 0.500 + 0.865i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.342300949\)
\(L(\frac12)\) \(\approx\) \(2.342300949\)
\(L(\frac{7}{2})\) 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 \( 1 + (-28.0 - 48.3i)T \)
23 \( 1 + 529iT \)
good3 \( 1 + 5.96iT - 243T^{2} \)
7 \( 1 + 183. iT - 1.68e4T^{2} \)
11 \( 1 - 76.8T + 1.61e5T^{2} \)
13 \( 1 - 418. iT - 3.71e5T^{2} \)
17 \( 1 - 1.76e3iT - 1.41e6T^{2} \)
19 \( 1 - 408.T + 2.47e6T^{2} \)
29 \( 1 - 4.71e3T + 2.05e7T^{2} \)
31 \( 1 - 5.67e3T + 2.86e7T^{2} \)
37 \( 1 - 4.28e3iT - 6.93e7T^{2} \)
41 \( 1 - 1.14e4T + 1.15e8T^{2} \)
43 \( 1 + 2.08e4iT - 1.47e8T^{2} \)
47 \( 1 - 1.88e4iT - 2.29e8T^{2} \)
53 \( 1 + 4.00e4iT - 4.18e8T^{2} \)
59 \( 1 - 2.54e4T + 7.14e8T^{2} \)
61 \( 1 + 2.80e4T + 8.44e8T^{2} \)
67 \( 1 + 7.08e4iT - 1.35e9T^{2} \)
71 \( 1 - 2.30e4T + 1.80e9T^{2} \)
73 \( 1 - 2.66e4iT - 2.07e9T^{2} \)
79 \( 1 + 208.T + 3.07e9T^{2} \)
83 \( 1 - 5.61e4iT - 3.93e9T^{2} \)
89 \( 1 - 2.01e4T + 5.58e9T^{2} \)
97 \( 1 - 7.95e3iT - 8.58e9T^{2} \)
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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

−10.94309409023881142594135584516, −10.33930606048362086017273097367, −9.663895693995072458176518501165, −8.140299324577306323350845963181, −7.04547389447417278446198923156, −6.32420641472656166056158612343, −4.47356984073453853777632055017, −3.58007335323008041451205428749, −2.01693270051394204203473670618, −0.976367436790554544145949178644, 0.969230865152590365300013487750, 2.69652280451941412235099733977, 4.52393562064174605371053780907, 5.28418872525294342833213522349, 6.19994468292241515649925878371, 7.57330890662642243218364592639, 8.706659121224079117035528844888, 9.374214414273681827723209259330, 10.11883405803919842354558507469, 11.74976496625925691500645460265

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