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Properties

Label 2-210-105.17-c2-0-19
Degree $2$
Conductor $210$
Sign $0.969 - 0.244i$
Analytic cond. $5.72208$
Root an. cond. $2.39208$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.36 + 0.366i)2-s + (2.39 − 1.80i)3-s + (1.73 + i)4-s + (−2.39 + 4.39i)5-s + (3.93 − 1.58i)6-s + (6.65 + 2.17i)7-s + (1.99 + 2i)8-s + (2.51 − 8.64i)9-s + (−4.87 + 5.12i)10-s + (5.82 + 3.36i)11-s + (5.95 − 0.718i)12-s + (−3.78 − 3.78i)13-s + (8.28 + 5.41i)14-s + (2.17 + 14.8i)15-s + (1.99 + 3.46i)16-s + (−3.69 + 0.990i)17-s + ⋯
L(s)  = 1  + (0.683 + 0.183i)2-s + (0.799 − 0.600i)3-s + (0.433 + 0.250i)4-s + (−0.478 + 0.878i)5-s + (0.656 − 0.263i)6-s + (0.950 + 0.311i)7-s + (0.249 + 0.250i)8-s + (0.279 − 0.960i)9-s + (−0.487 + 0.512i)10-s + (0.529 + 0.305i)11-s + (0.496 − 0.0598i)12-s + (−0.291 − 0.291i)13-s + (0.592 + 0.386i)14-s + (0.144 + 0.989i)15-s + (0.124 + 0.216i)16-s + (−0.217 + 0.0582i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 - 0.244i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.969 - 0.244i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.969 - 0.244i$
Analytic conductor: \(5.72208\)
Root analytic conductor: \(2.39208\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1),\ 0.969 - 0.244i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.89462 + 0.359393i\)
\(L(\frac12)\) \(\approx\) \(2.89462 + 0.359393i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.36 - 0.366i)T \)
3 \( 1 + (-2.39 + 1.80i)T \)
5 \( 1 + (2.39 - 4.39i)T \)
7 \( 1 + (-6.65 - 2.17i)T \)
good11 \( 1 + (-5.82 - 3.36i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (3.78 + 3.78i)T + 169iT^{2} \)
17 \( 1 + (3.69 - 0.990i)T + (250. - 144.5i)T^{2} \)
19 \( 1 + (-2.17 - 3.76i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (-3.48 + 12.9i)T + (-458. - 264.5i)T^{2} \)
29 \( 1 + 19.0T + 841T^{2} \)
31 \( 1 + (-0.140 - 0.0813i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-13.9 + 52.1i)T + (-1.18e3 - 684.5i)T^{2} \)
41 \( 1 + 64.3T + 1.68e3T^{2} \)
43 \( 1 + (49.3 - 49.3i)T - 1.84e3iT^{2} \)
47 \( 1 + (8.05 - 30.0i)T + (-1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (-79.5 + 21.3i)T + (2.43e3 - 1.40e3i)T^{2} \)
59 \( 1 + (-1.10 - 0.636i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (88.4 - 51.0i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (98.3 - 26.3i)T + (3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + 119. iT - 5.04e3T^{2} \)
73 \( 1 + (68.2 - 18.2i)T + (4.61e3 - 2.66e3i)T^{2} \)
79 \( 1 + (-89.1 + 51.4i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-21.3 + 21.3i)T - 6.88e3iT^{2} \)
89 \( 1 + (-83.4 + 48.1i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (37.6 - 37.6i)T - 9.40e3iT^{2} \)
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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

−12.16586329166763587043085159926, −11.56654234246018215406393698312, −10.40238353578975608209931801537, −8.944549572551757483530185117999, −7.87304550927945576934437222353, −7.19614121349204597102954802889, −6.11417516402510811558530418938, −4.49525485140075004678621130705, −3.26459477354597604724745606086, −1.99862739224900297722596807161, 1.68645398600756896642441860063, 3.50734632481813901198524214643, 4.49277335252147629513176852422, 5.27017219744100042069146147686, 7.16008506257621295331261470195, 8.224177401538587189260609942358, 9.042542119850768379499096640521, 10.19800152652265734578032264364, 11.36769636865594799288725775856, 11.98000580258568431608715048381

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