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Properties

Label 2-1412-1412.1175-c0-0-0
Degree $2$
Conductor $1412$
Sign $0.541 + 0.840i$
Analytic cond. $0.704679$
Root an. cond. $0.839452$
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  + i·2-s − 4-s + (−1.70 + 0.707i)5-s − i·8-s + (−0.707 + 0.707i)9-s + (−0.707 − 1.70i)10-s + (−0.292 − 0.707i)13-s + 16-s − 2i·17-s + (−0.707 − 0.707i)18-s + (1.70 − 0.707i)20-s + (1.70 − 1.70i)25-s + (0.707 − 0.292i)26-s + 1.41i·29-s + i·32-s + ⋯
L(s)  = 1  + i·2-s − 4-s + (−1.70 + 0.707i)5-s − i·8-s + (−0.707 + 0.707i)9-s + (−0.707 − 1.70i)10-s + (−0.292 − 0.707i)13-s + 16-s − 2i·17-s + (−0.707 − 0.707i)18-s + (1.70 − 0.707i)20-s + (1.70 − 1.70i)25-s + (0.707 − 0.292i)26-s + 1.41i·29-s + i·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1412 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.541 + 0.840i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1412 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.541 + 0.840i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1412\)    =    \(2^{2} \cdot 353\)
Sign: $0.541 + 0.840i$
Analytic conductor: \(0.704679\)
Root analytic conductor: \(0.839452\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1412} (1175, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1412,\ (\ :0),\ 0.541 + 0.840i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1885838676\)
\(L(\frac12)\) \(\approx\) \(0.1885838676\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
353 \( 1 - T \)
good3 \( 1 + (0.707 - 0.707i)T^{2} \)
5 \( 1 + (1.70 - 0.707i)T + (0.707 - 0.707i)T^{2} \)
7 \( 1 + (-0.707 - 0.707i)T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + (0.292 + 0.707i)T + (-0.707 + 0.707i)T^{2} \)
17 \( 1 + 2iT - T^{2} \)
19 \( 1 - iT^{2} \)
23 \( 1 - iT^{2} \)
29 \( 1 - 1.41iT - T^{2} \)
31 \( 1 + (-0.707 + 0.707i)T^{2} \)
37 \( 1 + (0.707 + 0.292i)T + (0.707 + 0.707i)T^{2} \)
41 \( 1 + (1.41 + 1.41i)T + iT^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + (1.70 + 0.707i)T + (0.707 + 0.707i)T^{2} \)
59 \( 1 + (-0.707 - 0.707i)T^{2} \)
61 \( 1 + 1.41iT - T^{2} \)
67 \( 1 + (0.707 - 0.707i)T^{2} \)
71 \( 1 + (0.707 + 0.707i)T^{2} \)
73 \( 1 + 1.41iT - T^{2} \)
79 \( 1 + (-0.707 - 0.707i)T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (-0.292 - 0.707i)T + (-0.707 + 0.707i)T^{2} \)
97 \( 1 + 2T + T^{2} \)
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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

−9.352239306634475146227615925769, −8.501289332187083766814094998508, −7.84318317388705677493747790394, −7.27153478713308962937103163255, −6.69594218822689271055234094185, −5.27637327241630881688371651239, −4.84503809474090266942637144101, −3.58437679196549262779861398294, −2.91721353694808270780910348619, −0.16457774914978776170939297267, 1.44321508084447495478673747456, 3.02665554166800284348553649559, 3.95847818690178995222865459080, 4.33174945342557487273042823197, 5.48665927799219889499954503997, 6.62099113893622231504465991360, 7.943829068629372331436818037799, 8.359044043211014577643781441613, 8.978698948886048078921057805482, 9.909600233034667625750234040246

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