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Properties

Label 2-430-215.49-c1-0-3
Degree $2$
Conductor $430$
Sign $-0.221 - 0.975i$
Analytic cond. $3.43356$
Root an. cond. $1.85298$
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 + (0.0417 + 0.0241i)3-s − 4-s + (0.704 + 2.12i)5-s + (0.0241 − 0.0417i)6-s + (−3.51 + 2.02i)7-s + i·8-s + (−1.49 − 2.59i)9-s + (2.12 − 0.704i)10-s + 1.21·11-s + (−0.0417 − 0.0241i)12-s + (−5.67 + 3.27i)13-s + (2.02 + 3.51i)14-s + (−0.0217 + 0.105i)15-s + 16-s + (0.427 − 0.246i)17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.0241 + 0.0139i)3-s − 0.5·4-s + (0.314 + 0.949i)5-s + (0.00984 − 0.0170i)6-s + (−1.32 + 0.766i)7-s + 0.353i·8-s + (−0.499 − 0.865i)9-s + (0.671 − 0.222i)10-s + 0.365·11-s + (−0.0120 − 0.00696i)12-s + (−1.57 + 0.908i)13-s + (0.541 + 0.938i)14-s + (−0.00561 + 0.0272i)15-s + 0.250·16-s + (0.103 − 0.0598i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.221 - 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.221 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(430\)    =    \(2 \cdot 5 \cdot 43\)
Sign: $-0.221 - 0.975i$
Analytic conductor: \(3.43356\)
Root analytic conductor: \(1.85298\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{430} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 430,\ (\ :1/2),\ -0.221 - 0.975i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.355412 + 0.445096i\)
\(L(\frac12)\) \(\approx\) \(0.355412 + 0.445096i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.704 - 2.12i)T \)
43 \( 1 + (2.56 - 6.03i)T \)
good3 \( 1 + (-0.0417 - 0.0241i)T + (1.5 + 2.59i)T^{2} \)
7 \( 1 + (3.51 - 2.02i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 - 1.21T + 11T^{2} \)
13 \( 1 + (5.67 - 3.27i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.427 + 0.246i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.95 - 5.12i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.64 + 2.10i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.43 - 2.47i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.02 + 6.97i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-7.39 - 4.26i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 8.65T + 41T^{2} \)
47 \( 1 - 3.61iT - 47T^{2} \)
53 \( 1 + (-3.83 - 2.21i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + 15.0T + 59T^{2} \)
61 \( 1 + (0.0822 + 0.142i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (10.8 + 6.28i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-0.973 - 1.68i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-9.35 + 5.40i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.639 - 1.10i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.265 + 0.153i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-2.37 + 4.11i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 15.1iT - 97T^{2} \)
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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

−11.62694966307951299597860403430, −10.35635771370822801703110606325, −9.565772881726402862454035501729, −9.274427418350172158118609395965, −7.78087365007690207584119488720, −6.34091414713773053983411814285, −6.12256903982940958960194878777, −4.28940519647942149297988822698, −3.09696141160079660340751697579, −2.33042784343099025395179741463, 0.33624415966601718602030942189, 2.67816224322288897872543890701, 4.26388306579502706782664658728, 5.18642086589659096753949467987, 6.16372355128689582522613691921, 7.22469165226407652800814608673, 8.042095337410139587656487664182, 9.108295471342798815252467012368, 9.842893448154400596450067983430, 10.58445058464257956829309266051

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