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Properties

Label 2-1305-435.434-c0-0-3
Degree $2$
Conductor $1305$
Sign $0.169 - 0.985i$
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·2-s + (0.707 + 0.707i)5-s − i·7-s + i·8-s + (−0.707 + 0.707i)10-s + 11-s − i·13-s + 14-s − 16-s − i·17-s + 1.41i·19-s + i·22-s + 1.00i·25-s + 26-s − 29-s + ⋯
L(s)  = 1  + i·2-s + (0.707 + 0.707i)5-s − i·7-s + i·8-s + (−0.707 + 0.707i)10-s + 11-s − i·13-s + 14-s − 16-s − i·17-s + 1.41i·19-s + i·22-s + 1.00i·25-s + 26-s − 29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.169 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.169 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1305\)    =    \(3^{2} \cdot 5 \cdot 29\)
Sign: $0.169 - 0.985i$
Analytic conductor: \(0.651279\)
Root analytic conductor: \(0.807019\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1305} (1304, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1305,\ (\ :0),\ 0.169 - 0.985i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.384129482\)
\(L(\frac12)\) \(\approx\) \(1.384129482\)
\(L(1)\) 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 \( 1 + (-0.707 - 0.707i)T \)
29 \( 1 + T \)
good2 \( 1 - iT - T^{2} \)
7 \( 1 + iT - T^{2} \)
11 \( 1 - T + T^{2} \)
13 \( 1 + iT - T^{2} \)
17 \( 1 + iT - T^{2} \)
19 \( 1 - 1.41iT - T^{2} \)
23 \( 1 + T^{2} \)
31 \( 1 + 1.41iT - T^{2} \)
37 \( 1 + 1.41T + T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 - iT - T^{2} \)
53 \( 1 + 1.41T + T^{2} \)
59 \( 1 - 1.41iT - T^{2} \)
61 \( 1 + 1.41iT - T^{2} \)
67 \( 1 - iT - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + T + T^{2} \)
97 \( 1 - 1.41T + T^{2} \)
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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

−9.951341626290097909224608454913, −9.240714301375710485092467306224, −8.036741708688563176233231622346, −7.43599191633021730782861685397, −6.79702715156859495192709331649, −5.99960822400291924195681016217, −5.38250658370185532485174349822, −4.05344393099094602045311876221, −3.00698395384819701814263092080, −1.67038125642667276524652896145, 1.53299647145158683911279144529, 2.12252140003167944483305309268, 3.35522843904284125642139184055, 4.39497443868652757035883322767, 5.37591952760100189981128218137, 6.41102835994116180479947010341, 6.93685517473482952822409025129, 8.536126068260713750722347587293, 9.098126507622570628551320832310, 9.534675365412083283193699606776

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