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Properties

Label 2-210-105.23-c1-0-2
Degree $2$
Conductor $210$
Sign $0.797 - 0.602i$
Analytic cond. $1.67685$
Root an. cond. $1.29493$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.965 + 0.258i)2-s + (−0.755 + 1.55i)3-s + (0.866 − 0.499i)4-s + (1.36 − 1.77i)5-s + (0.325 − 1.70i)6-s + (2.62 + 0.301i)7-s + (−0.707 + 0.707i)8-s + (−1.85 − 2.35i)9-s + (−0.854 + 2.06i)10-s + (1.49 − 0.860i)11-s + (0.125 + 1.72i)12-s + (1.71 + 1.71i)13-s + (−2.61 + 0.389i)14-s + (1.73 + 3.46i)15-s + (0.500 − 0.866i)16-s + (−0.641 + 2.39i)17-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + (−0.435 + 0.899i)3-s + (0.433 − 0.249i)4-s + (0.608 − 0.793i)5-s + (0.133 − 0.694i)6-s + (0.993 + 0.113i)7-s + (−0.249 + 0.249i)8-s + (−0.619 − 0.784i)9-s + (−0.270 + 0.653i)10-s + (0.449 − 0.259i)11-s + (0.0362 + 0.498i)12-s + (0.474 + 0.474i)13-s + (−0.699 + 0.104i)14-s + (0.449 + 0.893i)15-s + (0.125 − 0.216i)16-s + (−0.155 + 0.580i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.797 - 0.602i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.797 - 0.602i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.797 - 0.602i$
Analytic conductor: \(1.67685\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1/2),\ 0.797 - 0.602i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.899625 + 0.301558i\)
\(L(\frac12)\) \(\approx\) \(0.899625 + 0.301558i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 - 0.258i)T \)
3 \( 1 + (0.755 - 1.55i)T \)
5 \( 1 + (-1.36 + 1.77i)T \)
7 \( 1 + (-2.62 - 0.301i)T \)
good11 \( 1 + (-1.49 + 0.860i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.71 - 1.71i)T + 13iT^{2} \)
17 \( 1 + (0.641 - 2.39i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-3.74 - 2.16i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.97 - 7.36i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + 3.49T + 29T^{2} \)
31 \( 1 + (2.74 + 4.75i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.29 + 4.83i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 5.76iT - 41T^{2} \)
43 \( 1 + (1.33 + 1.33i)T + 43iT^{2} \)
47 \( 1 + (-5.84 + 1.56i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (13.1 + 3.51i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-5.45 - 9.44i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.22 - 7.31i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (13.2 + 3.54i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 10.9iT - 71T^{2} \)
73 \( 1 + (0.849 - 3.17i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-2.97 - 1.71i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.66 + 1.66i)T - 83iT^{2} \)
89 \( 1 + (-1.63 + 2.83i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-11.1 + 11.1i)T - 97iT^{2} \)
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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

−12.05886734176542195216398486942, −11.35922968651004100233684405324, −10.46346001256721974940977023340, −9.284684767429464123338491892813, −8.918751593707834840184584352840, −7.64478493685625655810246920814, −5.98086940080852276019750455481, −5.30598959592604776534200494962, −3.93052762802676076344489108597, −1.53733345438835020371749686408, 1.42379333889767310520947007921, 2.81097808396945860033262036521, 5.07744668633572782205183564050, 6.39292197539176836377405206469, 7.20068464640065200009923571160, 8.150748907669193161984735987945, 9.304518826703308038594586965669, 10.63357964625794000463404042291, 11.13987905361555925016644589158, 12.03406258289160007798633856367

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