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Properties

Label 2-10-5.3-c16-0-7
Degree $2$
Conductor $10$
Sign $-0.964 - 0.265i$
Analytic cond. $16.2324$
Root an. cond. $4.02895$
Motivic weight $16$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (128 − 128i)2-s + (−5.46e3 − 5.46e3i)3-s − 3.27e4i·4-s + (1.87e5 − 3.42e5i)5-s − 1.39e6·6-s + (6.16e6 − 6.16e6i)7-s + (−4.19e6 − 4.19e6i)8-s + 1.66e7i·9-s + (−1.98e7 − 6.78e7i)10-s − 3.83e8·11-s + (−1.79e8 + 1.79e8i)12-s + (6.97e8 + 6.97e8i)13-s − 1.57e9i·14-s + (−2.89e9 + 8.49e8i)15-s − 1.07e9·16-s + (2.15e9 − 2.15e9i)17-s + ⋯
L(s)  = 1  + (0.5 − 0.5i)2-s + (−0.832 − 0.832i)3-s − 0.5i·4-s + (0.479 − 0.877i)5-s − 0.832·6-s + (1.06 − 1.06i)7-s + (−0.250 − 0.250i)8-s + 0.387i·9-s + (−0.198 − 0.678i)10-s − 1.78·11-s + (−0.416 + 0.416i)12-s + (0.855 + 0.855i)13-s − 1.06i·14-s + (−1.13 + 0.331i)15-s − 0.250·16-s + (0.309 − 0.309i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.964 - 0.265i)\, \overline{\Lambda}(17-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+8) \, L(s)\cr =\mathstrut & (-0.964 - 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10\)    =    \(2 \cdot 5\)
Sign: $-0.964 - 0.265i$
Analytic conductor: \(16.2324\)
Root analytic conductor: \(4.02895\)
Motivic weight: \(16\)
Rational: no
Arithmetic: yes
Character: $\chi_{10} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 10,\ (\ :8),\ -0.964 - 0.265i)\)

Particular Values

\(L(\frac{17}{2})\) \(\approx\) \(0.227704 + 1.68661i\)
\(L(\frac12)\) \(\approx\) \(0.227704 + 1.68661i\)
\(L(9)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-128 + 128i)T \)
5 \( 1 + (-1.87e5 + 3.42e5i)T \)
good3 \( 1 + (5.46e3 + 5.46e3i)T + 4.30e7iT^{2} \)
7 \( 1 + (-6.16e6 + 6.16e6i)T - 3.32e13iT^{2} \)
11 \( 1 + 3.83e8T + 4.59e16T^{2} \)
13 \( 1 + (-6.97e8 - 6.97e8i)T + 6.65e17iT^{2} \)
17 \( 1 + (-2.15e9 + 2.15e9i)T - 4.86e19iT^{2} \)
19 \( 1 - 2.01e10iT - 2.88e20T^{2} \)
23 \( 1 + (3.86e10 + 3.86e10i)T + 6.13e21iT^{2} \)
29 \( 1 + 2.47e11iT - 2.50e23T^{2} \)
31 \( 1 - 1.30e12T + 7.27e23T^{2} \)
37 \( 1 + (-2.17e12 + 2.17e12i)T - 1.23e25iT^{2} \)
41 \( 1 - 1.11e12T + 6.37e25T^{2} \)
43 \( 1 + (9.81e11 + 9.81e11i)T + 1.36e26iT^{2} \)
47 \( 1 + (9.89e12 - 9.89e12i)T - 5.66e26iT^{2} \)
53 \( 1 + (-1.15e13 - 1.15e13i)T + 3.87e27iT^{2} \)
59 \( 1 + 1.15e14iT - 2.15e28T^{2} \)
61 \( 1 - 1.18e12T + 3.67e28T^{2} \)
67 \( 1 + (6.21e13 - 6.21e13i)T - 1.64e29iT^{2} \)
71 \( 1 + 7.46e14T + 4.16e29T^{2} \)
73 \( 1 + (1.01e14 + 1.01e14i)T + 6.50e29iT^{2} \)
79 \( 1 + 3.00e15iT - 2.30e30T^{2} \)
83 \( 1 + (5.42e13 + 5.42e13i)T + 5.07e30iT^{2} \)
89 \( 1 + 1.91e14iT - 1.54e31T^{2} \)
97 \( 1 + (-5.22e15 + 5.22e15i)T - 6.14e31iT^{2} \)
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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

−16.30071396701011062350596169343, −13.97738000557521218074102994759, −13.01611943239339410217488021497, −11.72004024167899632521356474022, −10.37220929706993103005886452958, −7.915328625662820699286656965218, −5.92002399075215450416092380850, −4.55326489844866103433342749139, −1.72694004599098914317528746125, −0.63603055830966978235801454657, 2.68156723862323460591152561473, 5.02365389683294656353045800049, 5.82837090438904012125288883424, 8.068861467221435393028483917405, 10.38557487219771977644631215453, 11.42814476696772686872153719490, 13.39754864380792556458198877287, 15.15679636710587472347765120953, 15.74349574435470919280105691853, 17.61183691069617713813937086327

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