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Properties

Label 2-1450-29.28-c1-0-7
Degree $2$
Conductor $1450$
Sign $-0.0908 - 0.995i$
Analytic cond. $11.5783$
Root an. cond. $3.40269$
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 − 2.06i·3-s − 4-s + 2.06·6-s + 1.81·7-s − i·8-s − 1.24·9-s + 3.98i·11-s + 2.06i·12-s − 6.04·13-s + 1.81i·14-s + 16-s + 6.58i·17-s − 1.24i·18-s + 4.06i·19-s + ⋯
L(s)  = 1  + 0.707i·2-s − 1.19i·3-s − 0.5·4-s + 0.841·6-s + 0.685·7-s − 0.353i·8-s − 0.416·9-s + 1.20i·11-s + 0.595i·12-s − 1.67·13-s + 0.484i·14-s + 0.250·16-s + 1.59i·17-s − 0.294i·18-s + 0.931i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0908 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0908 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1450\)    =    \(2 \cdot 5^{2} \cdot 29\)
Sign: $-0.0908 - 0.995i$
Analytic conductor: \(11.5783\)
Root analytic conductor: \(3.40269\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1450} (1101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1450,\ (\ :1/2),\ -0.0908 - 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.165804819\)
\(L(\frac12)\) \(\approx\) \(1.165804819\)
\(L(\frac{3}{2})\) 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 \( 1 \)
29 \( 1 + (5.36 - 0.489i)T \)
good3 \( 1 + 2.06iT - 3T^{2} \)
7 \( 1 - 1.81T + 7T^{2} \)
11 \( 1 - 3.98iT - 11T^{2} \)
13 \( 1 + 6.04T + 13T^{2} \)
17 \( 1 - 6.58iT - 17T^{2} \)
19 \( 1 - 4.06iT - 19T^{2} \)
23 \( 1 - 0.790T + 23T^{2} \)
31 \( 1 - 3.24iT - 31T^{2} \)
37 \( 1 + 0.457iT - 37T^{2} \)
41 \( 1 - 9.91iT - 41T^{2} \)
43 \( 1 + 3.62iT - 43T^{2} \)
47 \( 1 + 1.35iT - 47T^{2} \)
53 \( 1 - 6.90T + 53T^{2} \)
59 \( 1 - 14.5T + 59T^{2} \)
61 \( 1 - 0.827iT - 61T^{2} \)
67 \( 1 - 2.93T + 67T^{2} \)
71 \( 1 + 12.2T + 71T^{2} \)
73 \( 1 - 6.29iT - 73T^{2} \)
79 \( 1 - 13.9iT - 79T^{2} \)
83 \( 1 - 12.9T + 83T^{2} \)
89 \( 1 - 1.82iT - 89T^{2} \)
97 \( 1 + 17.0iT - 97T^{2} \)
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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

−9.782093141699180351019720970812, −8.535941940524637184736537256346, −7.897873414812702493080833403856, −7.30212005180113085681529082099, −6.75960470223348879653516376206, −5.74675057599852489509467704261, −4.87967206545641792138735068410, −3.98162032668657239854109221025, −2.26060921143968585208054709411, −1.48963265879860835317156496477, 0.46815423324071915231852133463, 2.31397466474804699543620850341, 3.18574664984685122045530970402, 4.22507987593542214596800653723, 5.00758668889841132612150680818, 5.43936193387590857425050649486, 7.03841153210742596454162792959, 7.82017104303032754434770569797, 9.013939279542201304070091996648, 9.312423753618871402265346378351

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