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Properties

Label 2-10-5.4-c25-0-4
Degree $2$
Conductor $10$
Sign $-0.923 - 0.383i$
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 + (−2.09e8 + 5.04e8i)5-s + 4.45e9·6-s + 4.89e10i·7-s + 6.87e10i·8-s − 3.34e11·9-s + (2.06e12 + 8.58e11i)10-s + 1.14e13·11-s − 1.82e13i·12-s + 1.28e14i·13-s + 2.00e14·14-s + (−5.47e14 − 2.27e14i)15-s + 2.81e14·16-s + 3.78e15i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + 1.18i·3-s − 0.5·4-s + (−0.383 + 0.923i)5-s + 0.835·6-s + 1.33i·7-s + 0.353i·8-s − 0.394·9-s + (0.652 + 0.271i)10-s + 1.10·11-s − 0.590i·12-s + 1.52i·13-s + 0.945·14-s + (−1.09 − 0.453i)15-s + 0.250·16-s + 1.57i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 - 0.383i)\, \overline{\Lambda}(26-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+25/2) \, L(s)\cr =\mathstrut & (-0.923 - 0.383i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10\)    =    \(2 \cdot 5\)
Sign: $-0.923 - 0.383i$
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: \((2,\ 10,\ (\ :25/2),\ -0.923 - 0.383i)\)

Particular Values

\(L(13)\) \(\approx\) \(1.736292756\)
\(L(\frac12)\) \(\approx\) \(1.736292756\)
\(L(\frac{27}{2})\) 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 \( 1 + (2.09e8 - 5.04e8i)T \)
good3 \( 1 - 1.08e6iT - 8.47e11T^{2} \)
7 \( 1 - 4.89e10iT - 1.34e21T^{2} \)
11 \( 1 - 1.14e13T + 1.08e26T^{2} \)
13 \( 1 - 1.28e14iT - 7.05e27T^{2} \)
17 \( 1 - 3.78e15iT - 5.77e30T^{2} \)
19 \( 1 - 1.02e16T + 9.30e31T^{2} \)
23 \( 1 + 1.81e17iT - 1.10e34T^{2} \)
29 \( 1 + 2.19e18T + 3.63e36T^{2} \)
31 \( 1 - 2.89e18T + 1.92e37T^{2} \)
37 \( 1 - 4.56e19iT - 1.60e39T^{2} \)
41 \( 1 + 4.22e18T + 2.08e40T^{2} \)
43 \( 1 - 1.22e19iT - 6.86e40T^{2} \)
47 \( 1 - 4.60e20iT - 6.34e41T^{2} \)
53 \( 1 + 2.54e21iT - 1.27e43T^{2} \)
59 \( 1 - 2.20e21T + 1.86e44T^{2} \)
61 \( 1 - 7.41e21T + 4.29e44T^{2} \)
67 \( 1 + 6.01e22iT - 4.48e45T^{2} \)
71 \( 1 - 3.59e22T + 1.91e46T^{2} \)
73 \( 1 + 1.98e23iT - 3.82e46T^{2} \)
79 \( 1 - 7.84e23T + 2.75e47T^{2} \)
83 \( 1 + 4.60e23iT - 9.48e47T^{2} \)
89 \( 1 + 5.94e23T + 5.42e48T^{2} \)
97 \( 1 + 8.71e24iT - 4.66e49T^{2} \)
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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

−15.20091443597688812358307381776, −14.40609180095253654814629851564, −12.12553904510257564857110198575, −11.15828913378639232702672048351, −9.801434825655895616201000220696, −8.742854435306203558544389162421, −6.33060047635509501127809841407, −4.45444854442630657853519353231, −3.42023671624776729608996587988, −1.92829890306561245638169520314, 0.61107436700693429559351187088, 1.11650476265713127741070073633, 3.73930868362870722723133905791, 5.38951166417021480369549848168, 7.20673926993925637102437086111, 7.72493197197274905152943347575, 9.506616825257490031707822571769, 11.74664333832208700734733677894, 13.12703234702800444727696547278, 13.88168160011231437970817076402

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