Properties

Label 2-1400-5.4-c1-0-25
Degree $2$
Conductor $1400$
Sign $-0.894 - 0.447i$
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  − 1.56i·3-s i·7-s + 0.561·9-s − 6.12·11-s + 2i·13-s − 1.56i·17-s − 3.56·19-s − 1.56·21-s + 1.43i·23-s − 5.56i·27-s − 3.43·29-s − 9.12·31-s + 9.56i·33-s − 8.80i·37-s + 3.12·39-s + ⋯
L(s)  = 1  − 0.901i·3-s − 0.377i·7-s + 0.187·9-s − 1.84·11-s + 0.554i·13-s − 0.378i·17-s − 0.817·19-s − 0.340·21-s + 0.299i·23-s − 1.07i·27-s − 0.638·29-s − 1.63·31-s + 1.66i·33-s − 1.44i·37-s + 0.500·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $-0.894 - 0.447i$
Analytic conductor: \(11.1790\)
Root analytic conductor: \(3.34350\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1400,\ (\ :1/2),\ -0.894 - 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3259281401\)
\(L(\frac12)\) \(\approx\) \(0.3259281401\)
\(L(\frac{3}{2})\) 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 \( 1 + iT \)
good3 \( 1 + 1.56iT - 3T^{2} \)
11 \( 1 + 6.12T + 11T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
17 \( 1 + 1.56iT - 17T^{2} \)
19 \( 1 + 3.56T + 19T^{2} \)
23 \( 1 - 1.43iT - 23T^{2} \)
29 \( 1 + 3.43T + 29T^{2} \)
31 \( 1 + 9.12T + 31T^{2} \)
37 \( 1 + 8.80iT - 37T^{2} \)
41 \( 1 + 2.43T + 41T^{2} \)
43 \( 1 - 6.56iT - 43T^{2} \)
47 \( 1 - 8.24iT - 47T^{2} \)
53 \( 1 + 1.12iT - 53T^{2} \)
59 \( 1 + 11.3T + 59T^{2} \)
61 \( 1 - 11.1T + 61T^{2} \)
67 \( 1 + 7.87iT - 67T^{2} \)
71 \( 1 - 1.68T + 71T^{2} \)
73 \( 1 - 6.43iT - 73T^{2} \)
79 \( 1 - 5.68T + 79T^{2} \)
83 \( 1 + 1.31iT - 83T^{2} \)
89 \( 1 + 9.80T + 89T^{2} \)
97 \( 1 + 6iT - 97T^{2} \)
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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

−9.081162752673009206428389654400, −7.958717293194539767873055845748, −7.55280305967758657100233381963, −6.83488907603964730613411195985, −5.85005395599561985619280843546, −4.95880589178186559231077555954, −3.91982692896788961672814128614, −2.60554095707743777150431627647, −1.71510285824784588129406347453, −0.12143691650617283800001316686, 2.03334467768135975053386425005, 3.12731312007437915747506060590, 4.08372290730380335181854380122, 5.15063957064894969698554384189, 5.52212033311335163036353834290, 6.79253430005078459046597179521, 7.76356082406297453785215464323, 8.454093945827793902222504359180, 9.287417716803989372260488826399, 10.29400094772509713632163057182

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