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Properties

Label 2-20e2-100.23-c1-0-6
Degree $2$
Conductor $400$
Sign $0.911 - 0.411i$
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  + (1.56 + 1.59i)5-s + (1.76 − 2.42i)9-s + (0.552 + 3.48i)13-s + (2.87 + 1.46i)17-s + (−0.0759 + 4.99i)25-s + (2.51 + 0.816i)29-s + (11.3 − 1.80i)37-s + (−9.69 − 7.04i)41-s + (6.63 − 0.999i)45-s − 7i·49-s + (−8.83 + 4.50i)53-s + (−4.69 + 3.41i)61-s + (−4.68 + 6.35i)65-s + (−16.7 − 2.65i)73-s + (−2.78 − 8.55i)81-s + ⋯
L(s)  = 1  + (0.701 + 0.712i)5-s + (0.587 − 0.809i)9-s + (0.153 + 0.967i)13-s + (0.697 + 0.355i)17-s + (−0.0151 + 0.999i)25-s + (0.466 + 0.151i)29-s + (1.87 − 0.296i)37-s + (−1.51 − 1.09i)41-s + (0.988 − 0.148i)45-s − i·49-s + (−1.21 + 0.618i)53-s + (−0.601 + 0.436i)61-s + (−0.581 + 0.787i)65-s + (−1.96 − 0.310i)73-s + (−0.309 − 0.951i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.911 - 0.411i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.911 - 0.411i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $0.911 - 0.411i$
Analytic conductor: \(3.19401\)
Root analytic conductor: \(1.78718\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :1/2),\ 0.911 - 0.411i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.56848 + 0.338117i\)
\(L(\frac12)\) \(\approx\) \(1.56848 + 0.338117i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.56 - 1.59i)T \)
good3 \( 1 + (-1.76 + 2.42i)T^{2} \)
7 \( 1 + 7iT^{2} \)
11 \( 1 + (-3.39 + 10.4i)T^{2} \)
13 \( 1 + (-0.552 - 3.48i)T + (-12.3 + 4.01i)T^{2} \)
17 \( 1 + (-2.87 - 1.46i)T + (9.99 + 13.7i)T^{2} \)
19 \( 1 + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-21.8 - 7.10i)T^{2} \)
29 \( 1 + (-2.51 - 0.816i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (25.0 - 18.2i)T^{2} \)
37 \( 1 + (-11.3 + 1.80i)T + (35.1 - 11.4i)T^{2} \)
41 \( 1 + (9.69 + 7.04i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 43iT^{2} \)
47 \( 1 + (27.6 - 38.0i)T^{2} \)
53 \( 1 + (8.83 - 4.50i)T + (31.1 - 42.8i)T^{2} \)
59 \( 1 + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (4.69 - 3.41i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-39.3 - 54.2i)T^{2} \)
71 \( 1 + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (16.7 + 2.65i)T + (69.4 + 22.5i)T^{2} \)
79 \( 1 + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (48.7 + 67.1i)T^{2} \)
89 \( 1 + (11.0 + 15.2i)T + (-27.5 + 84.6i)T^{2} \)
97 \( 1 + (2.30 + 4.53i)T + (-57.0 + 78.4i)T^{2} \)
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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

−11.31846659876304025506936557229, −10.27820935762468373574552825494, −9.656419556897359043035691832673, −8.760048442304330771539653475765, −7.39860692192092177774708143310, −6.59429791978219530152749638918, −5.78621671371601767291501765341, −4.32403245741819042049033184615, −3.15076108754147102243304003941, −1.64174447385110877099949727392, 1.32498208675624670537217557117, 2.84144497269799488259998536440, 4.50983919418615539013356299270, 5.33200494608851954996447901833, 6.33298906173385501211470046378, 7.69574265544466145156000085377, 8.344221863687439031530016149718, 9.607677667182834172014289712764, 10.12061502948257167724389714767, 11.12461586266970147187250463007

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