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Properties

Label 2-513-171.122-c1-0-8
Degree $2$
Conductor $513$
Sign $0.460 + 0.887i$
Analytic cond. $4.09632$
Root an. cond. $2.02393$
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  − 2.58·2-s + 4.66·4-s + (1.09 − 0.629i)5-s + (0.0839 + 0.145i)7-s − 6.89·8-s + (−2.81 + 1.62i)10-s + (3.58 − 2.07i)11-s − 5.36i·13-s + (−0.216 − 0.375i)14-s + 8.46·16-s + (−0.632 − 0.364i)17-s + (0.391 + 4.34i)19-s + (5.09 − 2.94i)20-s + (−9.26 + 5.35i)22-s − 4.93i·23-s + ⋯
L(s)  = 1  − 1.82·2-s + 2.33·4-s + (0.487 − 0.281i)5-s + (0.0317 + 0.0549i)7-s − 2.43·8-s + (−0.890 + 0.514i)10-s + (1.08 − 0.624i)11-s − 1.48i·13-s + (−0.0579 − 0.100i)14-s + 2.11·16-s + (−0.153 − 0.0885i)17-s + (0.0897 + 0.995i)19-s + (1.13 − 0.657i)20-s + (−1.97 + 1.14i)22-s − 1.02i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.460 + 0.887i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.460 + 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(513\)    =    \(3^{3} \cdot 19\)
Sign: $0.460 + 0.887i$
Analytic conductor: \(4.09632\)
Root analytic conductor: \(2.02393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{513} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 513,\ (\ :1/2),\ 0.460 + 0.887i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.569993 - 0.346320i\)
\(L(\frac12)\) \(\approx\) \(0.569993 - 0.346320i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
19 \( 1 + (-0.391 - 4.34i)T \)
good2 \( 1 + 2.58T + 2T^{2} \)
5 \( 1 + (-1.09 + 0.629i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (-0.0839 - 0.145i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-3.58 + 2.07i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 5.36iT - 13T^{2} \)
17 \( 1 + (0.632 + 0.364i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + 4.93iT - 23T^{2} \)
29 \( 1 + (2.05 - 3.56i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.96 + 2.86i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 8.98iT - 37T^{2} \)
41 \( 1 + (-3.69 - 6.40i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 11.1T + 43T^{2} \)
47 \( 1 + (1.95 + 1.13i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.41 + 5.90i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.693 - 1.20i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.23 + 5.59i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + 8.03iT - 67T^{2} \)
71 \( 1 + (-6.11 + 10.5i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (1.47 - 2.54i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 1.03iT - 79T^{2} \)
83 \( 1 + (-7.02 + 4.05i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (0.114 + 0.197i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 18.4iT - 97T^{2} \)
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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

−10.63125569246897888357928487659, −9.603985498586581982771807577657, −9.096427223959760986021720672811, −8.208781351966657505447943207797, −7.48883454952845757525393953443, −6.33040122073410436494112075874, −5.57346295327550923758664694546, −3.52800022281757022157536383594, −2.06625301967356248583744693769, −0.77153376877955486880888652488, 1.43242320168513947978298641768, 2.42175600732633709589005070576, 4.20144694568377928061917045064, 6.01445713054637952456494656963, 6.88513583175258371587881514888, 7.40220958625549434179860019584, 8.709479889007961507632159634779, 9.362887503543894935954694678227, 9.773651825742971079160512004833, 10.90772293851199965838275499388

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