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Properties

Label 2-3200-40.13-c0-0-1
Degree $2$
Conductor $3200$
Sign $-0.973 - 0.229i$
Analytic cond. $1.59700$
Root an. cond. $1.26372$
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  + (−1.41 + 1.41i)7-s + i·9-s + (−1.41 − 1.41i)23-s − 2·41-s + (−1.41 + 1.41i)47-s − 3.00i·49-s + (−1.41 − 1.41i)63-s − 81-s + 2i·89-s + (1.41 + 1.41i)103-s + ⋯
L(s)  = 1  + (−1.41 + 1.41i)7-s + i·9-s + (−1.41 − 1.41i)23-s − 2·41-s + (−1.41 + 1.41i)47-s − 3.00i·49-s + (−1.41 − 1.41i)63-s − 81-s + 2i·89-s + (1.41 + 1.41i)103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3200\)    =    \(2^{7} \cdot 5^{2}\)
Sign: $-0.973 - 0.229i$
Analytic conductor: \(1.59700\)
Root analytic conductor: \(1.26372\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3200} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3200,\ (\ :0),\ -0.973 - 0.229i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4347297935\)
\(L(\frac12)\) \(\approx\) \(0.4347297935\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−9.148326532307363632626620187203, −8.456745219485019832428474431609, −7.893583112563146795830117800031, −6.69503468329614598464856202422, −6.26241272749764046438391669309, −5.46083967693233716720656857596, −4.69652154913408887139214636706, −3.54382545189060467677852852507, −2.67783678238172333166591511944, −1.99642760280359098530146809054, 0.24654595920807105591038488511, 1.62408912239523120760558423085, 3.28993053408557069037493646195, 3.55069198096291171232947161275, 4.40442741252701725036055526623, 5.63017924837728775204799129432, 6.41473516732947081626218693252, 6.91657645981859781069903239951, 7.59710677473257601591282565970, 8.533938438053315476957157777162

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