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Properties

Label 2-1400-35.4-c1-0-1
Degree $2$
Conductor $1400$
Sign $-0.995 - 0.0902i$
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  + (1.09 + 0.632i)3-s + (−2.16 − 1.51i)7-s + (−0.699 − 1.21i)9-s + (−3.21 + 5.57i)11-s + 1.10i·13-s + (3.89 + 2.25i)17-s + (−2.93 − 5.08i)19-s + (−1.41 − 3.03i)21-s + (−7.59 + 4.38i)23-s − 5.56i·27-s − 3.87·29-s + (3.33 − 5.77i)31-s + (−7.04 + 4.06i)33-s + (−1.23 + 0.713i)37-s + (−0.698 + 1.21i)39-s + ⋯
L(s)  = 1  + (0.632 + 0.365i)3-s + (−0.819 − 0.573i)7-s + (−0.233 − 0.403i)9-s + (−0.969 + 1.67i)11-s + 0.306i·13-s + (0.945 + 0.546i)17-s + (−0.673 − 1.16i)19-s + (−0.308 − 0.662i)21-s + (−1.58 + 0.914i)23-s − 1.07i·27-s − 0.719·29-s + (0.598 − 1.03i)31-s + (−1.22 + 0.708i)33-s + (−0.203 + 0.117i)37-s + (−0.111 + 0.193i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0902i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 - 0.0902i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $-0.995 - 0.0902i$
Analytic conductor: \(11.1790\)
Root analytic conductor: \(3.34350\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (249, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1400,\ (\ :1/2),\ -0.995 - 0.0902i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3097510163\)
\(L(\frac12)\) \(\approx\) \(0.3097510163\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.16 + 1.51i)T \)
good3 \( 1 + (-1.09 - 0.632i)T + (1.5 + 2.59i)T^{2} \)
11 \( 1 + (3.21 - 5.57i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 1.10iT - 13T^{2} \)
17 \( 1 + (-3.89 - 2.25i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.93 + 5.08i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (7.59 - 4.38i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 3.87T + 29T^{2} \)
31 \( 1 + (-3.33 + 5.77i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.23 - 0.713i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 11.8T + 41T^{2} \)
43 \( 1 - 6.46iT - 43T^{2} \)
47 \( 1 + (7.32 - 4.23i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.665 - 0.384i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.898 - 1.55i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.51 + 2.63i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.47 - 1.42i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 9.47T + 71T^{2} \)
73 \( 1 + (-11.6 - 6.69i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.65 + 8.05i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 3.20iT - 83T^{2} \)
89 \( 1 + (1.46 + 2.53i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 5.96iT - 97T^{2} \)
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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

−9.809612686518908545009598492341, −9.470114953556537436648310682833, −8.276020415955115341961864952181, −7.62729439760536446709004177173, −6.77379475767612281238953208532, −5.90927007622293912874005088406, −4.69099519084187318150675327699, −3.92712241369976550872649192060, −3.01779048242196887607937939639, −1.94375256429330830274065448983, 0.10392839459534479723910115571, 2.00281856183551238410903277817, 3.02467918429917658516279745963, 3.55579421606432951507185359421, 5.26243910530463369277756188050, 5.78898034750065032815162523470, 6.67427606613997750086464409163, 7.935637208747163621138832334491, 8.263755413373597965299673493007, 8.897052164308464196354845238091

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