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Properties

Label 2-1300-13.3-c1-0-21
Degree $2$
Conductor $1300$
Sign $0.0128 - 0.999i$
Analytic cond. $10.3805$
Root an. cond. $3.22188$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.5 − 2.59i)3-s + (1.5 − 2.59i)7-s + (−3 + 5.19i)9-s + (−1.5 − 2.59i)11-s + (−1 + 3.46i)13-s + (−3.5 + 6.06i)17-s + (−0.5 + 0.866i)19-s − 9·21-s + (−3.5 − 6.06i)23-s + 9·27-s + (2.5 + 4.33i)29-s − 4·31-s + (−4.5 + 7.79i)33-s + (−1.5 − 2.59i)37-s + (10.5 − 2.59i)39-s + ⋯
L(s)  = 1  + (−0.866 − 1.49i)3-s + (0.566 − 0.981i)7-s + (−1 + 1.73i)9-s + (−0.452 − 0.783i)11-s + (−0.277 + 0.960i)13-s + (−0.848 + 1.47i)17-s + (−0.114 + 0.198i)19-s − 1.96·21-s + (−0.729 − 1.26i)23-s + 1.73·27-s + (0.464 + 0.804i)29-s − 0.718·31-s + (−0.783 + 1.35i)33-s + (−0.246 − 0.427i)37-s + (1.68 − 0.416i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0128 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0128 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1300\)    =    \(2^{2} \cdot 5^{2} \cdot 13\)
Sign: $0.0128 - 0.999i$
Analytic conductor: \(10.3805\)
Root analytic conductor: \(3.22188\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1300} (601, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 1300,\ (\ :1/2),\ 0.0128 - 0.999i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
13 \( 1 + (1 - 3.46i)T \)
good3 \( 1 + (1.5 + 2.59i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (-1.5 + 2.59i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (3.5 - 6.06i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.5 + 6.06i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.5 - 4.33i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + (1.5 + 2.59i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.5 + 6.06i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.5 - 7.79i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 8T + 47T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + (2.5 - 4.33i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.5 + 4.33i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.5 - 11.2i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.5 + 2.59i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 14T + 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 + 12T + 83T^{2} \)
89 \( 1 + (3.5 + 6.06i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.5 - 9.52i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.585969607847656533658149962535, −8.188866215337405605822046758518, −7.20703242500657490431464843555, −6.63827870737620377859315036970, −5.95544377306830087584443184215, −4.90840778938318118168275529986, −3.91584942792265496549680569467, −2.22026313476912934380673198913, −1.36121347330447919011366207862, 0, 2.27776506567585466296082140876, 3.42719028933260261725774967244, 4.64531064442203995783209427605, 5.13469758025166394851909168936, 5.68599325105049423049565254550, 6.85058468830513833592092932877, 7.970190154733630184024690287172, 8.869055039101236915360092102865, 9.771859769313799426295011235013

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