Properties

Label 2-1450-5.4-c1-0-35
Degree $2$
Conductor $1450$
Sign $-0.447 + 0.894i$
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 − 2.44i·3-s − 4-s + 2.44·6-s − 0.449i·7-s i·8-s − 2.99·9-s + 2·11-s + 2.44i·12-s − 2.44i·13-s + 0.449·14-s + 16-s − 2i·17-s − 2.99i·18-s − 1.55·19-s + ⋯
L(s)  = 1  + 0.707i·2-s − 1.41i·3-s − 0.5·4-s + 0.999·6-s − 0.169i·7-s − 0.353i·8-s − 0.999·9-s + 0.603·11-s + 0.707i·12-s − 0.679i·13-s + 0.120·14-s + 0.250·16-s − 0.485i·17-s − 0.707i·18-s − 0.355·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1450\)    =    \(2 \cdot 5^{2} \cdot 29\)
Sign: $-0.447 + 0.894i$
Analytic conductor: \(11.5783\)
Root analytic conductor: \(3.40269\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1450} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1450,\ (\ :1/2),\ -0.447 + 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.211932376\)
\(L(\frac12)\) \(\approx\) \(1.211932376\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
good3 \( 1 + 2.44iT - 3T^{2} \)
7 \( 1 + 0.449iT - 7T^{2} \)
11 \( 1 - 2T + 11T^{2} \)
13 \( 1 + 2.44iT - 13T^{2} \)
17 \( 1 + 2iT - 17T^{2} \)
19 \( 1 + 1.55T + 19T^{2} \)
23 \( 1 + 2.44iT - 23T^{2} \)
31 \( 1 + 3T + 31T^{2} \)
37 \( 1 - 1.44iT - 37T^{2} \)
41 \( 1 + 3.34T + 41T^{2} \)
43 \( 1 + 0.898iT - 43T^{2} \)
47 \( 1 + 3.89iT - 47T^{2} \)
53 \( 1 + 5.55iT - 53T^{2} \)
59 \( 1 + 13.4T + 59T^{2} \)
61 \( 1 + 1.44T + 61T^{2} \)
67 \( 1 + 8.55iT - 67T^{2} \)
71 \( 1 + 3.34T + 71T^{2} \)
73 \( 1 - 1.34iT - 73T^{2} \)
79 \( 1 + 6.89T + 79T^{2} \)
83 \( 1 - 6iT - 83T^{2} \)
89 \( 1 + 8.44T + 89T^{2} \)
97 \( 1 + 7.34iT - 97T^{2} \)
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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

−8.931133470799966285867921755611, −8.242251604341794977444603964977, −7.51096503197804036898234892367, −6.87463634680763686940395310552, −6.25297418006538971067670274510, −5.38659916503155576644925348815, −4.27614888635459448993043043867, −3.02016948314528278590651837676, −1.72554745935703660930138230652, −0.49700477550509189015436159945, 1.62013578213626474716980281381, 2.96375911662350427182263275907, 3.92776772645798855927164009089, 4.40907723764914993691765018579, 5.37360136234552349302611584651, 6.29222070660684011500727322197, 7.49679986597995208862348698193, 8.720333017252885705808512429768, 9.119575156057513304525276921743, 9.824782191007785247335856860283

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