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Properties

Label 2-1300-65.4-c1-0-10
Degree $2$
Conductor $1300$
Sign $0.796 - 0.604i$
Analytic cond. $10.3805$
Root an. cond. $3.22188$
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  + (2.01 + 1.16i)3-s + (−0.199 − 0.346i)7-s + (1.21 + 2.11i)9-s + (1.5 + 0.866i)11-s + (3.55 − 0.619i)13-s + (0.599 − 0.346i)17-s + (4.65 − 2.68i)19-s − 0.932i·21-s + (−0.0927 − 0.0535i)23-s − 1.30i·27-s + (−2.45 + 4.24i)29-s + 7.86i·31-s + (2.01 + 3.49i)33-s + (1.13 − 1.96i)37-s + (7.89 + 2.89i)39-s + ⋯
L(s)  = 1  + (1.16 + 0.673i)3-s + (−0.0755 − 0.130i)7-s + (0.406 + 0.704i)9-s + (0.452 + 0.261i)11-s + (0.985 − 0.171i)13-s + (0.145 − 0.0839i)17-s + (1.06 − 0.616i)19-s − 0.203i·21-s + (−0.0193 − 0.0111i)23-s − 0.251i·27-s + (−0.455 + 0.788i)29-s + 1.41i·31-s + (0.351 + 0.608i)33-s + (0.186 − 0.322i)37-s + (1.26 + 0.462i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.796 - 0.604i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.796 - 0.604i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1300\)    =    \(2^{2} \cdot 5^{2} \cdot 13\)
Sign: $0.796 - 0.604i$
Analytic conductor: \(10.3805\)
Root analytic conductor: \(3.22188\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1300} (849, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1300,\ (\ :1/2),\ 0.796 - 0.604i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.668315941\)
\(L(\frac12)\) \(\approx\) \(2.668315941\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.55 + 0.619i)T \)
good3 \( 1 + (-2.01 - 1.16i)T + (1.5 + 2.59i)T^{2} \)
7 \( 1 + (0.199 + 0.346i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.5 - 0.866i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-0.599 + 0.346i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.65 + 2.68i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.0927 + 0.0535i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.45 - 4.24i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 7.86iT - 31T^{2} \)
37 \( 1 + (-1.13 + 1.96i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-6.69 - 3.86i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.20 - 3.00i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 3.46T + 47T^{2} \)
53 \( 1 - 11.7iT - 53T^{2} \)
59 \( 1 + (-6.30 + 3.63i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.34 + 7.52i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.664 - 1.15i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.35 - 1.93i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 10.2T + 73T^{2} \)
79 \( 1 - 13.1T + 79T^{2} \)
83 \( 1 + 14.0T + 83T^{2} \)
89 \( 1 + (0.300 + 0.173i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-4.42 - 7.66i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.515341054944514488189256567771, −9.022639761070797069466552201922, −8.314911126310566578724855540095, −7.44353719933754312789337218034, −6.54759319402972909797243642789, −5.40413733570314540006830745573, −4.39028660043460177332560569534, −3.50078295266069475087571036813, −2.86936397822286558976000450909, −1.37720020819409994578030191554, 1.20973498535056029290590929124, 2.28066942926131942216011803759, 3.34507639153333173416903151526, 4.06161960614182141331593598376, 5.56856281619861505635592551050, 6.32304681198048245001564282549, 7.37611797861150667063661894792, 7.946210326975140153520307366356, 8.710893461169134562887841361793, 9.336056779955214904585102736032

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