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Properties

Label 2-10-5.4-c25-0-8
Degree $2$
Conductor $10$
Sign $-0.832 + 0.554i$
Analytic cond. $39.5996$
Root an. cond. $6.29282$
Motivic weight $25$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.09e3i·2-s − 1.08e6i·3-s − 1.67e7·4-s + (3.02e8 + 4.54e8i)5-s − 4.42e9·6-s + 5.43e10i·7-s + 6.87e10i·8-s − 3.19e11·9-s + (1.86e12 − 1.23e12i)10-s − 3.24e12·11-s + 1.81e13i·12-s − 7.83e13i·13-s + 2.22e14·14-s + (4.90e14 − 3.26e14i)15-s + 2.81e14·16-s − 4.33e15i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s − 1.17i·3-s − 0.5·4-s + (0.554 + 0.832i)5-s − 0.829·6-s + 1.48i·7-s + 0.353i·8-s − 0.377·9-s + (0.588 − 0.391i)10-s − 0.311·11-s + 0.586i·12-s − 0.932i·13-s + 1.04·14-s + (0.976 − 0.650i)15-s + 0.250·16-s − 1.80i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 + 0.554i)\, \overline{\Lambda}(26-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+25/2) \, L(s)\cr =\mathstrut & (-0.832 + 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10\)    =    \(2 \cdot 5\)
Sign: $-0.832 + 0.554i$
Analytic conductor: \(39.5996\)
Root analytic conductor: \(6.29282\)
Motivic weight: \(25\)
Rational: no
Arithmetic: yes
Character: $\chi_{10} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 10,\ (\ :25/2),\ -0.832 + 0.554i)\)

Particular Values

\(L(13)\) \(\approx\) \(1.680922401\)
\(L(\frac12)\) \(\approx\) \(1.680922401\)
\(L(\frac{27}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 4.09e3iT \)
5 \( 1 + (-3.02e8 - 4.54e8i)T \)
good3 \( 1 + 1.08e6iT - 8.47e11T^{2} \)
7 \( 1 - 5.43e10iT - 1.34e21T^{2} \)
11 \( 1 + 3.24e12T + 1.08e26T^{2} \)
13 \( 1 + 7.83e13iT - 7.05e27T^{2} \)
17 \( 1 + 4.33e15iT - 5.77e30T^{2} \)
19 \( 1 + 8.87e15T + 9.30e31T^{2} \)
23 \( 1 + 7.05e16iT - 1.10e34T^{2} \)
29 \( 1 - 1.11e18T + 3.63e36T^{2} \)
31 \( 1 - 6.30e18T + 1.92e37T^{2} \)
37 \( 1 + 1.55e19iT - 1.60e39T^{2} \)
41 \( 1 + 3.82e19T + 2.08e40T^{2} \)
43 \( 1 + 3.32e20iT - 6.86e40T^{2} \)
47 \( 1 + 1.01e21iT - 6.34e41T^{2} \)
53 \( 1 + 3.43e21iT - 1.27e43T^{2} \)
59 \( 1 + 4.09e21T + 1.86e44T^{2} \)
61 \( 1 - 3.14e22T + 4.29e44T^{2} \)
67 \( 1 - 1.01e23iT - 4.48e45T^{2} \)
71 \( 1 - 2.10e23T + 1.91e46T^{2} \)
73 \( 1 + 1.56e23iT - 3.82e46T^{2} \)
79 \( 1 + 6.00e22T + 2.75e47T^{2} \)
83 \( 1 - 7.40e23iT - 9.48e47T^{2} \)
89 \( 1 + 3.21e24T + 5.42e48T^{2} \)
97 \( 1 + 5.32e24iT - 4.66e49T^{2} \)
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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

−13.87236572299090109891194606974, −12.71424536878652317712436764746, −11.67962470663177634298721341658, −10.05603040562248664153373248341, −8.428486773459191756039542905798, −6.77030513099262456585126097058, −5.39715016337200091938986744390, −2.73035937751645888533419825930, −2.21208734750881956932416112428, −0.51440054404701933124201165982, 1.27066684019661583230237795593, 4.00810229415146463704744948073, 4.66340108303199837851136257039, 6.37287996471786296848069900468, 8.225330184264824418588629141151, 9.658921550965568600161136106927, 10.59936336221846833356843329490, 12.98613512955200363180129285159, 14.17397780615487213232238113067, 15.60792001111095505592873150970

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