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Properties

Label 2-10-5.4-c7-0-3
Degree $2$
Conductor $10$
Sign $-0.669 + 0.743i$
Analytic cond. $3.12385$
Root an. cond. $1.76744$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 8i·2-s − 27.6i·3-s − 64·4-s + (−207. − 187. i)5-s − 221.·6-s − 593. i·7-s + 512i·8-s + 1.42e3·9-s + (−1.49e3 + 1.66e3i)10-s + 5.34e3·11-s + 1.77e3i·12-s + 2.59e3i·13-s − 4.74e3·14-s + (−5.17e3 + 5.74e3i)15-s + 4.09e3·16-s − 3.11e4i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.591i·3-s − 0.5·4-s + (−0.743 − 0.669i)5-s − 0.418·6-s − 0.654i·7-s + 0.353i·8-s + 0.649·9-s + (−0.473 + 0.525i)10-s + 1.21·11-s + 0.295i·12-s + 0.327i·13-s − 0.462·14-s + (−0.396 + 0.439i)15-s + 0.250·16-s − 1.53i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.669 + 0.743i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.669 + 0.743i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10\)    =    \(2 \cdot 5\)
Sign: $-0.669 + 0.743i$
Analytic conductor: \(3.12385\)
Root analytic conductor: \(1.76744\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{10} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 10,\ (\ :7/2),\ -0.669 + 0.743i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.484153 - 1.08746i\)
\(L(\frac12)\) \(\approx\) \(0.484153 - 1.08746i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 8iT \)
5 \( 1 + (207. + 187. i)T \)
good3 \( 1 + 27.6iT - 2.18e3T^{2} \)
7 \( 1 + 593. iT - 8.23e5T^{2} \)
11 \( 1 - 5.34e3T + 1.94e7T^{2} \)
13 \( 1 - 2.59e3iT - 6.27e7T^{2} \)
17 \( 1 + 3.11e4iT - 4.10e8T^{2} \)
19 \( 1 + 1.88e4T + 8.93e8T^{2} \)
23 \( 1 - 8.30e4iT - 3.40e9T^{2} \)
29 \( 1 - 1.86e5T + 1.72e10T^{2} \)
31 \( 1 + 1.68e5T + 2.75e10T^{2} \)
37 \( 1 + 3.44e5iT - 9.49e10T^{2} \)
41 \( 1 - 4.90e5T + 1.94e11T^{2} \)
43 \( 1 - 1.01e5iT - 2.71e11T^{2} \)
47 \( 1 - 1.39e6iT - 5.06e11T^{2} \)
53 \( 1 + 2.94e5iT - 1.17e12T^{2} \)
59 \( 1 - 1.18e6T + 2.48e12T^{2} \)
61 \( 1 + 1.23e6T + 3.14e12T^{2} \)
67 \( 1 - 9.28e5iT - 6.06e12T^{2} \)
71 \( 1 - 1.00e6T + 9.09e12T^{2} \)
73 \( 1 - 2.48e6iT - 1.10e13T^{2} \)
79 \( 1 + 8.42e5T + 1.92e13T^{2} \)
83 \( 1 + 4.01e6iT - 2.71e13T^{2} \)
89 \( 1 + 1.03e7T + 4.42e13T^{2} \)
97 \( 1 - 9.39e4iT - 8.07e13T^{2} \)
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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

−19.19153721801846597138172749033, −17.61829497038603593608639454115, −16.10674329362955484033028275404, −13.99706425165724811076311947306, −12.60460992918555797720767998348, −11.43367107693412124962151191428, −9.317644992569716089361416705973, −7.33587238188566509601062922084, −4.20656131698610987747333475998, −1.05704594893896707814521876076, 4.09086863770160314906664794995, 6.56402003137759792453762556057, 8.557909576579256318904402123345, 10.45349184246193126009731803838, 12.39358215853472067479845814737, 14.66043325761920922479715697619, 15.37232311014966958122166066029, 16.74450824033852987675823636175, 18.40702188714093166165563582801, 19.59516531522986997785847322382

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