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Properties

Label 2-1305-5.4-c1-0-18
Degree $2$
Conductor $1305$
Sign $-0.931 + 0.364i$
Analytic cond. $10.4204$
Root an. cond. $3.22807$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.51i·2-s − 4.34·4-s + (2.08 − 0.814i)5-s + 4.08i·7-s − 5.91i·8-s + (2.05 + 5.24i)10-s + 1.07·11-s − 1.43i·13-s − 10.2·14-s + 6.20·16-s + 6.88i·17-s + 7.42·19-s + (−9.05 + 3.54i)20-s + 2.71i·22-s + 1.01i·23-s + ⋯
L(s)  = 1  + 1.78i·2-s − 2.17·4-s + (0.931 − 0.364i)5-s + 1.54i·7-s − 2.09i·8-s + (0.649 + 1.65i)10-s + 0.325·11-s − 0.397i·13-s − 2.74·14-s + 1.55·16-s + 1.67i·17-s + 1.70·19-s + (−2.02 + 0.791i)20-s + 0.579i·22-s + 0.212i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.931 + 0.364i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.931 + 0.364i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1305\)    =    \(3^{2} \cdot 5 \cdot 29\)
Sign: $-0.931 + 0.364i$
Analytic conductor: \(10.4204\)
Root analytic conductor: \(3.22807\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1305} (784, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1305,\ (\ :1/2),\ -0.931 + 0.364i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.620458967\)
\(L(\frac12)\) \(\approx\) \(1.620458967\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.08 + 0.814i)T \)
29 \( 1 - T \)
good2 \( 1 - 2.51iT - 2T^{2} \)
7 \( 1 - 4.08iT - 7T^{2} \)
11 \( 1 - 1.07T + 11T^{2} \)
13 \( 1 + 1.43iT - 13T^{2} \)
17 \( 1 - 6.88iT - 17T^{2} \)
19 \( 1 - 7.42T + 19T^{2} \)
23 \( 1 - 1.01iT - 23T^{2} \)
31 \( 1 + 7.37T + 31T^{2} \)
37 \( 1 - 10.8iT - 37T^{2} \)
41 \( 1 + 3.02T + 41T^{2} \)
43 \( 1 + 10.6iT - 43T^{2} \)
47 \( 1 - 6.88iT - 47T^{2} \)
53 \( 1 - 7.61iT - 53T^{2} \)
59 \( 1 + 9.45T + 59T^{2} \)
61 \( 1 + 0.265T + 61T^{2} \)
67 \( 1 + 11.1iT - 67T^{2} \)
71 \( 1 + 12.1T + 71T^{2} \)
73 \( 1 + 3.54iT - 73T^{2} \)
79 \( 1 - 6.13T + 79T^{2} \)
83 \( 1 - 0.615iT - 83T^{2} \)
89 \( 1 - 2.11T + 89T^{2} \)
97 \( 1 + 1.52iT - 97T^{2} \)
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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

−9.531520998684073690100826952594, −9.087484710433083623701253389665, −8.434959191639255383241305622265, −7.69458349714275299455608582159, −6.59394642647028422326211969302, −5.83947800651882888836642908165, −5.54519709627304857263803364799, −4.69423810541791626909407414317, −3.23473275191521161443641162536, −1.64969312778584402888974133589, 0.72472375149696487557192936242, 1.70305422072896613229485385881, 2.89230156399119772700036415544, 3.63504052677711535567047755255, 4.62208312084623053947626986983, 5.46167964996784014629328450771, 6.92165489139613011561750555032, 7.48429613240855605811888393432, 8.991000517511363714967487429663, 9.596797303937147160553066960929

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