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Properties

Label 2-1400-35.9-c1-0-10
Degree $2$
Conductor $1400$
Sign $-0.489 - 0.872i$
Analytic cond. $11.1790$
Root an. cond. $3.34350$
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  + (−0.761 + 0.439i)3-s + (2.27 + 1.35i)7-s + (−1.11 + 1.92i)9-s + (1.59 + 2.75i)11-s + 0.490i·13-s + (0.829 − 0.479i)17-s + (−1.05 + 1.82i)19-s + (−2.32 − 0.0362i)21-s + (−5.44 − 3.14i)23-s − 4.59i·27-s − 0.588·29-s + (3.73 + 6.47i)31-s + (−2.42 − 1.40i)33-s + (−1.30 − 0.754i)37-s + (−0.215 − 0.373i)39-s + ⋯
L(s)  = 1  + (−0.439 + 0.253i)3-s + (0.858 + 0.513i)7-s + (−0.371 + 0.642i)9-s + (0.480 + 0.831i)11-s + 0.135i·13-s + (0.201 − 0.116i)17-s + (−0.241 + 0.418i)19-s + (−0.507 − 0.00791i)21-s + (−1.13 − 0.655i)23-s − 0.884i·27-s − 0.109·29-s + (0.671 + 1.16i)31-s + (−0.422 − 0.243i)33-s + (−0.214 − 0.124i)37-s + (−0.0345 − 0.0597i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.489 - 0.872i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.489 - 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $-0.489 - 0.872i$
Analytic conductor: \(11.1790\)
Root analytic conductor: \(3.34350\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (849, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1400,\ (\ :1/2),\ -0.489 - 0.872i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.198998564\)
\(L(\frac12)\) \(\approx\) \(1.198998564\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 + (-2.27 - 1.35i)T \)
good3 \( 1 + (0.761 - 0.439i)T + (1.5 - 2.59i)T^{2} \)
11 \( 1 + (-1.59 - 2.75i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 0.490iT - 13T^{2} \)
17 \( 1 + (-0.829 + 0.479i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.05 - 1.82i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (5.44 + 3.14i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 0.588T + 29T^{2} \)
31 \( 1 + (-3.73 - 6.47i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.30 + 0.754i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 4.29T + 41T^{2} \)
43 \( 1 + 5.92iT - 43T^{2} \)
47 \( 1 + (-4.47 - 2.58i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (11.4 - 6.58i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.31 - 4.01i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.96 - 12.0i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.30 + 3.63i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 12.1T + 71T^{2} \)
73 \( 1 + (8.77 - 5.06i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.26 - 2.19i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 4.80iT - 83T^{2} \)
89 \( 1 + (5.40 - 9.36i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 10.5iT - 97T^{2} \)
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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

−9.989261047297940396259745166363, −8.935157511369694836225723822866, −8.263580398287773692727356271017, −7.49814279557919652366915703243, −6.44613933582527003545809929815, −5.58933772028664755581932783294, −4.81551318766659768684520362778, −4.10206787303586938250066634130, −2.58142770193775324446693150148, −1.61135379256950364304935676904, 0.52165761968286040169901395688, 1.73415244966530852164689536454, 3.26006544177251362366989330765, 4.13344238969609124014855796231, 5.20423094987560996883484654156, 6.06751373752726075599909653852, 6.69671051407151640848391756419, 7.80520726405052145100387982474, 8.339915892859347039554587284767, 9.307376956490306784778438012586

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