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Properties

Label 2-1400-35.9-c1-0-26
Degree $2$
Conductor $1400$
Sign $-0.652 + 0.758i$
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  + (−0.866 + 0.5i)3-s + (−2.59 − 0.5i)7-s + (−1 + 1.73i)9-s + (1 + 1.73i)11-s + 4i·13-s + (3 − 5.19i)19-s + (2.5 − 0.866i)21-s + (−2.59 − 1.5i)23-s − 5i·27-s + 3·29-s + (−1.73 − 0.999i)33-s + (−10.3 − 6i)37-s + (−2 − 3.46i)39-s − 7·41-s − 9i·43-s + ⋯
L(s)  = 1  + (−0.499 + 0.288i)3-s + (−0.981 − 0.188i)7-s + (−0.333 + 0.577i)9-s + (0.301 + 0.522i)11-s + 1.10i·13-s + (0.688 − 1.19i)19-s + (0.545 − 0.188i)21-s + (−0.541 − 0.312i)23-s − 0.962i·27-s + 0.557·29-s + (−0.301 − 0.174i)33-s + (−1.70 − 0.986i)37-s + (−0.320 − 0.554i)39-s − 1.09·41-s − 1.37i·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.652 + 0.758i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.652 + 0.758i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $-0.652 + 0.758i$
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: \((2,\ 1400,\ (\ :1/2),\ -0.652 + 0.758i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1635754625\)
\(L(\frac12)\) \(\approx\) \(0.1635754625\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.59 + 0.5i)T \)
good3 \( 1 + (0.866 - 0.5i)T + (1.5 - 2.59i)T^{2} \)
11 \( 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 4iT - 13T^{2} \)
17 \( 1 + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3 + 5.19i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.59 + 1.5i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 3T + 29T^{2} \)
31 \( 1 + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (10.3 + 6i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 7T + 41T^{2} \)
43 \( 1 + 9iT - 43T^{2} \)
47 \( 1 + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.19 - 3i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (5 + 8.66i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.5 - 4.33i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (9.52 - 5.5i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 10T + 71T^{2} \)
73 \( 1 + (6.92 - 4i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3 + 5.19i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 3iT - 83T^{2} \)
89 \( 1 + (-8.5 + 14.7i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 2iT - 97T^{2} \)
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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

−9.246790415538467373845830420765, −8.734786929011440242931760510145, −7.40397532552265046886276233144, −6.83078533784247317800508408900, −5.99713427961379273547716370704, −5.00970025529159122714410260969, −4.25683563651642515280401964899, −3.14883063946513827653123226655, −1.95411974131313597969562195462, −0.07322748617058085621290372614, 1.30801961132419506638461242743, 3.11786366517122804357721820476, 3.50118076867954482843762302973, 5.05394865355582098786243154960, 5.99768990619082366079163978303, 6.29147724460214324233297888925, 7.36181403041561759539957016470, 8.270074361513489325456464062623, 9.059741400201493415091559886666, 9.967553102986274695145946058595

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