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Properties

Label 2-20e2-80.69-c1-0-4
Degree $2$
Conductor $400$
Sign $0.816 + 0.577i$
Analytic cond. $3.19401$
Root an. cond. $1.78718$
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  + (−1.27 + 0.618i)2-s + (−2.16 − 2.16i)3-s + (1.23 − 1.57i)4-s + (4.09 + 1.41i)6-s + 3.30·7-s + (−0.594 + 2.76i)8-s + 6.40i·9-s + (2.01 + 2.01i)11-s + (−6.08 + 0.738i)12-s + (−0.794 − 0.794i)13-s + (−4.20 + 2.04i)14-s + (−0.955 − 3.88i)16-s + 4.61i·17-s + (−3.96 − 8.14i)18-s + (3.48 − 3.48i)19-s + ⋯
L(s)  = 1  + (−0.899 + 0.437i)2-s + (−1.25 − 1.25i)3-s + (0.616 − 0.787i)4-s + (1.67 + 0.577i)6-s + 1.24·7-s + (−0.210 + 0.977i)8-s + 2.13i·9-s + (0.606 + 0.606i)11-s + (−1.75 + 0.213i)12-s + (−0.220 − 0.220i)13-s + (−1.12 + 0.546i)14-s + (−0.238 − 0.971i)16-s + 1.11i·17-s + (−0.934 − 1.91i)18-s + (0.800 − 0.800i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $0.816 + 0.577i$
Analytic conductor: \(3.19401\)
Root analytic conductor: \(1.78718\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :1/2),\ 0.816 + 0.577i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.678297 - 0.215441i\)
\(L(\frac12)\) \(\approx\) \(0.678297 - 0.215441i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.27 - 0.618i)T \)
5 \( 1 \)
good3 \( 1 + (2.16 + 2.16i)T + 3iT^{2} \)
7 \( 1 - 3.30T + 7T^{2} \)
11 \( 1 + (-2.01 - 2.01i)T + 11iT^{2} \)
13 \( 1 + (0.794 + 0.794i)T + 13iT^{2} \)
17 \( 1 - 4.61iT - 17T^{2} \)
19 \( 1 + (-3.48 + 3.48i)T - 19iT^{2} \)
23 \( 1 - 7.99T + 23T^{2} \)
29 \( 1 + (-1.95 + 1.95i)T - 29iT^{2} \)
31 \( 1 + 5.12T + 31T^{2} \)
37 \( 1 + (0.448 - 0.448i)T - 37iT^{2} \)
41 \( 1 + 4.02iT - 41T^{2} \)
43 \( 1 + (-4.97 + 4.97i)T - 43iT^{2} \)
47 \( 1 + 5.49iT - 47T^{2} \)
53 \( 1 + (3.35 - 3.35i)T - 53iT^{2} \)
59 \( 1 + (2.07 + 2.07i)T + 59iT^{2} \)
61 \( 1 + (0.557 - 0.557i)T - 61iT^{2} \)
67 \( 1 + (0.636 + 0.636i)T + 67iT^{2} \)
71 \( 1 - 6.85iT - 71T^{2} \)
73 \( 1 - 10.5T + 73T^{2} \)
79 \( 1 - 17.3T + 79T^{2} \)
83 \( 1 + (9.48 + 9.48i)T + 83iT^{2} \)
89 \( 1 - 7.62iT - 89T^{2} \)
97 \( 1 - 0.709iT - 97T^{2} \)
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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

−11.11745872983162537697819348059, −10.62234371215263401096542439830, −9.201733789772059507405194580553, −8.137284474593260151801298094180, −7.31473452577066923159185938218, −6.75805805046265095866488604449, −5.60855509789479609525774999189, −4.89495198524986229897065022891, −2.01581368431498722039898824923, −1.02166949809489882922900829329, 1.09443395573418354072888547522, 3.28273609356675499785486346762, 4.53700305836783353917162457737, 5.40529041871930745725411843393, 6.65969908926893245032403741178, 7.79676028796335589393706145876, 9.076526994948635030262173580434, 9.549859517799528726269152564835, 10.68453066547990805689999609924, 11.24392205133731316298644145308

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