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Properties

Label 2-5-5.3-c18-0-1
Degree $2$
Conductor $5$
Sign $0.542 - 0.839i$
Analytic cond. $10.2693$
Root an. cond. $3.20457$
Motivic weight $18$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (367. − 367. i)2-s + (5.23e3 + 5.23e3i)3-s − 7.50e3i·4-s + (−3.98e4 + 1.95e6i)5-s + 3.84e6·6-s + (−4.50e7 + 4.50e7i)7-s + (9.35e7 + 9.35e7i)8-s − 3.32e8i·9-s + (7.02e8 + 7.31e8i)10-s + 3.12e8·11-s + (3.92e7 − 3.92e7i)12-s + (1.04e10 + 1.04e10i)13-s + 3.30e10i·14-s + (−1.04e10 + 1.00e10i)15-s + 7.06e10·16-s + (−1.10e10 + 1.10e10i)17-s + ⋯
L(s)  = 1  + (0.717 − 0.717i)2-s + (0.265 + 0.265i)3-s − 0.0286i·4-s + (−0.0204 + 0.999i)5-s + 0.381·6-s + (−1.11 + 1.11i)7-s + (0.696 + 0.696i)8-s − 0.858i·9-s + (0.702 + 0.731i)10-s + 0.132·11-s + (0.00760 − 0.00760i)12-s + (0.982 + 0.982i)13-s + 1.60i·14-s + (−0.271 + 0.260i)15-s + 1.02·16-s + (−0.0930 + 0.0930i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.542 - 0.839i)\, \overline{\Lambda}(19-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+9) \, L(s)\cr =\mathstrut & (0.542 - 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $0.542 - 0.839i$
Analytic conductor: \(10.2693\)
Root analytic conductor: \(3.20457\)
Motivic weight: \(18\)
Rational: no
Arithmetic: yes
Character: $\chi_{5} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5,\ (\ :9),\ 0.542 - 0.839i)\)

Particular Values

\(L(\frac{19}{2})\) \(\approx\) \(2.11694 + 1.15210i\)
\(L(\frac12)\) \(\approx\) \(2.11694 + 1.15210i\)
\(L(10)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (3.98e4 - 1.95e6i)T \)
good2 \( 1 + (-367. + 367. i)T - 2.62e5iT^{2} \)
3 \( 1 + (-5.23e3 - 5.23e3i)T + 3.87e8iT^{2} \)
7 \( 1 + (4.50e7 - 4.50e7i)T - 1.62e15iT^{2} \)
11 \( 1 - 3.12e8T + 5.55e18T^{2} \)
13 \( 1 + (-1.04e10 - 1.04e10i)T + 1.12e20iT^{2} \)
17 \( 1 + (1.10e10 - 1.10e10i)T - 1.40e22iT^{2} \)
19 \( 1 + 3.29e11iT - 1.04e23T^{2} \)
23 \( 1 + (6.53e11 + 6.53e11i)T + 3.24e24iT^{2} \)
29 \( 1 - 2.14e13iT - 2.10e26T^{2} \)
31 \( 1 - 8.16e11T + 6.99e26T^{2} \)
37 \( 1 + (-1.46e14 + 1.46e14i)T - 1.68e28iT^{2} \)
41 \( 1 - 1.45e14T + 1.07e29T^{2} \)
43 \( 1 + (-4.92e14 - 4.92e14i)T + 2.52e29iT^{2} \)
47 \( 1 + (-8.51e13 + 8.51e13i)T - 1.25e30iT^{2} \)
53 \( 1 + (-2.27e14 - 2.27e14i)T + 1.08e31iT^{2} \)
59 \( 1 + 1.24e16iT - 7.50e31T^{2} \)
61 \( 1 - 1.05e15T + 1.36e32T^{2} \)
67 \( 1 + (-2.81e16 + 2.81e16i)T - 7.40e32iT^{2} \)
71 \( 1 + 8.30e15T + 2.10e33T^{2} \)
73 \( 1 + (-5.67e15 - 5.67e15i)T + 3.46e33iT^{2} \)
79 \( 1 - 1.08e17iT - 1.43e34T^{2} \)
83 \( 1 + (-8.31e16 - 8.31e16i)T + 3.49e34iT^{2} \)
89 \( 1 + 2.93e17iT - 1.22e35T^{2} \)
97 \( 1 + (6.44e17 - 6.44e17i)T - 5.77e35iT^{2} \)
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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

−19.68022938343049226144503338106, −18.25313116699841385235882440989, −15.88214831283513202191528885068, −14.34610365615423457356698976843, −12.66460021924590564579451146458, −11.23940970423410650086726474140, −9.182743605540296652991102591649, −6.40692861503537236880815573691, −3.71703611017311163357340789018, −2.58637362347973051677985611966, 0.933202697606091261481187208019, 4.04044855090925861950481762622, 5.90519591985498768714035122361, 7.77789594586567095603156674467, 10.14872347136561845406428074945, 13.04997462960537982506778053496, 13.69780830311479457649056856438, 15.81854940058198923093094155683, 16.75648939701122086855604019919, 19.26156322945907718901873337178

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