Properties

Degree 2
Conductor $ 2^{2} \cdot 7 \cdot 11 \cdot 13 $
Sign $0.536 + 0.844i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.819·3-s + 1.69i·5-s + i·7-s − 2.32·9-s i·11-s + (−1.93 − 3.04i)13-s − 1.39i·15-s − 2.32·17-s + 2.77i·19-s − 0.819i·21-s − 0.366·23-s + 2.11·25-s + 4.36·27-s + 2.95·29-s + 9.83i·31-s + ⋯
L(s)  = 1  − 0.472·3-s + 0.759i·5-s + 0.377i·7-s − 0.776·9-s − 0.301i·11-s + (−0.536 − 0.844i)13-s − 0.358i·15-s − 0.562·17-s + 0.636i·19-s − 0.178i·21-s − 0.0763·23-s + 0.423·25-s + 0.840·27-s + 0.548·29-s + 1.76i·31-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.536 + 0.844i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.536 + 0.844i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4004\)    =    \(2^{2} \cdot 7 \cdot 11 \cdot 13\)
\( \varepsilon \)  =  $0.536 + 0.844i$
motivic weight  =  \(1\)
character  :  $\chi_{4004} (2157, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4004,\ (\ :1/2),\ 0.536 + 0.844i)$
$L(1)$  $\approx$  $0.7774422125$
$L(\frac12)$  $\approx$  $0.7774422125$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;7,\;11,\;13\}$, \(F_p\) is a polynomial of degree 2. If $p \in \{2,\;7,\;11,\;13\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
7 \( 1 - iT \)
11 \( 1 + iT \)
13 \( 1 + (1.93 + 3.04i)T \)
good3 \( 1 + 0.819T + 3T^{2} \)
5 \( 1 - 1.69iT - 5T^{2} \)
17 \( 1 + 2.32T + 17T^{2} \)
19 \( 1 - 2.77iT - 19T^{2} \)
23 \( 1 + 0.366T + 23T^{2} \)
29 \( 1 - 2.95T + 29T^{2} \)
31 \( 1 - 9.83iT - 31T^{2} \)
37 \( 1 + 7.39iT - 37T^{2} \)
41 \( 1 - 9.67iT - 41T^{2} \)
43 \( 1 + 10.1T + 43T^{2} \)
47 \( 1 + 10.4iT - 47T^{2} \)
53 \( 1 + 7.02T + 53T^{2} \)
59 \( 1 + 11.9iT - 59T^{2} \)
61 \( 1 + 8.21T + 61T^{2} \)
67 \( 1 + 10.3iT - 67T^{2} \)
71 \( 1 - 0.127iT - 71T^{2} \)
73 \( 1 + 11.1iT - 73T^{2} \)
79 \( 1 - 14.2T + 79T^{2} \)
83 \( 1 - 4.44iT - 83T^{2} \)
89 \( 1 + 5.15iT - 89T^{2} \)
97 \( 1 - 4.55iT - 97T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.312055019769829483929945089650, −7.64469915222158284246101055824, −6.54383645182702499981017899936, −6.34839711156367714421716604484, −5.25866286187437633845014345121, −4.87920729999505368645303046191, −3.36660237315337456981040297957, −3.00642607284053422454673108063, −1.88599821111770440670050877288, −0.30359305548731041263741442787, 0.843404690440819695236917679801, 2.08340287375729561695442129529, 3.04255983970197795013789766794, 4.36759142865176784664957838792, 4.67382457905137158805682880419, 5.54303439876884691476093735508, 6.38734372452644128277463798780, 6.99294709820353082407331268564, 7.87469100376566395819969773506, 8.653990869365176967578092155328

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