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Properties

Label 2-1305-145.28-c0-0-0
Degree $2$
Conductor $1305$
Sign $0.525 - 0.850i$
Analytic cond. $0.651279$
Root an. cond. $0.807019$
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·4-s + i·5-s + (−1 + i)7-s + (1 + i)13-s − 16-s + 20-s + (1 + i)23-s − 25-s + (1 + i)28-s + i·29-s + (−1 − i)35-s − i·49-s + (1 − i)52-s + (1 + i)53-s − 2i·59-s + ⋯
L(s)  = 1  − i·4-s + i·5-s + (−1 + i)7-s + (1 + i)13-s − 16-s + 20-s + (1 + i)23-s − 25-s + (1 + i)28-s + i·29-s + (−1 − i)35-s − i·49-s + (1 − i)52-s + (1 + i)53-s − 2i·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1305\)    =    \(3^{2} \cdot 5 \cdot 29\)
Sign: $0.525 - 0.850i$
Analytic conductor: \(0.651279\)
Root analytic conductor: \(0.807019\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1305} (28, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1305,\ (\ :0),\ 0.525 - 0.850i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9240038691\)
\(L(\frac12)\) \(\approx\) \(0.9240038691\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 - iT \)
29 \( 1 - iT \)
good2 \( 1 + iT^{2} \)
7 \( 1 + (1 - i)T - iT^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + (-1 - i)T + iT^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + (-1 - i)T + iT^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + (-1 - i)T + iT^{2} \)
59 \( 1 + 2iT - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + (-1 + i)T - iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + (1 + i)T + iT^{2} \)
89 \( 1 + T^{2} \)
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

−9.838494888648135015458781081418, −9.321189407297538903634678514303, −8.650729164496488770198983932516, −7.20836192124090762125305406272, −6.53920060225715955635184696216, −5.99638669612119762934698929537, −5.15327193117871385703819638045, −3.76001498082527382282625424920, −2.83925183425261035585244694899, −1.70273985209171435836332150353, 0.823605612749546619287545682450, 2.72896600927028467486062128253, 3.72951469275092571050344583144, 4.31278416644434131834467802188, 5.51342830395093572867571532635, 6.54658043699249328741430300525, 7.32418807031249877466122374417, 8.235167077384079541482623192683, 8.713964683257326198593180607220, 9.656032733823675555912549513925

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