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Properties

Label 2-44e2-44.15-c0-0-4
Degree $2$
Conductor $1936$
Sign $0.530 + 0.847i$
Analytic cond. $0.966189$
Root an. cond. $0.982949$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.61 − 1.17i)5-s + (−0.809 − 0.587i)9-s + (0.927 − 2.85i)25-s + (0.618 + 1.90i)37-s − 2·45-s + (−0.809 + 0.587i)49-s + (−1.61 − 1.17i)53-s + (0.309 + 0.951i)81-s + 2·89-s + (1.61 + 1.17i)97-s + (−0.618 + 1.90i)113-s + ⋯
L(s)  = 1  + (1.61 − 1.17i)5-s + (−0.809 − 0.587i)9-s + (0.927 − 2.85i)25-s + (0.618 + 1.90i)37-s − 2·45-s + (−0.809 + 0.587i)49-s + (−1.61 − 1.17i)53-s + (0.309 + 0.951i)81-s + 2·89-s + (1.61 + 1.17i)97-s + (−0.618 + 1.90i)113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.530 + 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.530 + 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1936\)    =    \(2^{4} \cdot 11^{2}\)
Sign: $0.530 + 0.847i$
Analytic conductor: \(0.966189\)
Root analytic conductor: \(0.982949\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1936} (1775, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1936,\ (\ :0),\ 0.530 + 0.847i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.425085024\)
\(L(\frac12)\) \(\approx\) \(1.425085024\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 \)
good3 \( 1 + (0.809 + 0.587i)T^{2} \)
5 \( 1 + (-1.61 + 1.17i)T + (0.309 - 0.951i)T^{2} \)
7 \( 1 + (0.809 - 0.587i)T^{2} \)
13 \( 1 + (0.309 + 0.951i)T^{2} \)
17 \( 1 + (0.309 - 0.951i)T^{2} \)
19 \( 1 + (0.809 + 0.587i)T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (-0.809 + 0.587i)T^{2} \)
31 \( 1 + (-0.309 - 0.951i)T^{2} \)
37 \( 1 + (-0.618 - 1.90i)T + (-0.809 + 0.587i)T^{2} \)
41 \( 1 + (-0.809 - 0.587i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.809 + 0.587i)T^{2} \)
53 \( 1 + (1.61 + 1.17i)T + (0.309 + 0.951i)T^{2} \)
59 \( 1 + (0.809 - 0.587i)T^{2} \)
61 \( 1 + (0.309 - 0.951i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (-0.309 + 0.951i)T^{2} \)
73 \( 1 + (-0.809 + 0.587i)T^{2} \)
79 \( 1 + (-0.309 - 0.951i)T^{2} \)
83 \( 1 + (-0.309 + 0.951i)T^{2} \)
89 \( 1 - 2T + T^{2} \)
97 \( 1 + (-1.61 - 1.17i)T + (0.309 + 0.951i)T^{2} \)
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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

−9.341118511736887463295921258058, −8.614597888452366576757973710693, −7.977904936158001137245768035883, −6.45762337152535848706026644966, −6.16751525478329651278275421926, −5.21161221062349062698801480406, −4.65808734410828753826416517646, −3.24126915348585452078129591020, −2.18490455960584567185170656406, −1.11010788754681871727495403493, 1.82944126276248611970861440434, 2.57329974492798289371383538778, 3.38536251320907929983731987403, 4.85973323335521222352093999177, 5.75848180293032728282588841598, 6.15748568949449400737520160146, 7.07458695754486411503440974634, 7.84302837399376757472193556393, 8.959133070245407718120480533255, 9.518848267459121723309916913396

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