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Properties

Label 2-210-15.2-c1-0-7
Degree $2$
Conductor $210$
Sign $0.378 + 0.925i$
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  + (−0.707 + 0.707i)2-s + (1.27 − 1.17i)3-s − 1.00i·4-s + (−1.37 − 1.76i)5-s + (−0.0727 + 1.73i)6-s + (−0.707 − 0.707i)7-s + (0.707 + 0.707i)8-s + (0.251 − 2.98i)9-s + (2.21 + 0.275i)10-s − 2.48i·11-s + (−1.17 − 1.27i)12-s + (−1.31 + 1.31i)13-s + 1.00·14-s + (−3.82 − 0.637i)15-s − 1.00·16-s + (2.15 − 2.15i)17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (0.736 − 0.676i)3-s − 0.500i·4-s + (−0.614 − 0.788i)5-s + (−0.0296 + 0.706i)6-s + (−0.267 − 0.267i)7-s + (0.250 + 0.250i)8-s + (0.0838 − 0.996i)9-s + (0.701 + 0.0869i)10-s − 0.750i·11-s + (−0.338 − 0.368i)12-s + (−0.363 + 0.363i)13-s + 0.267·14-s + (−0.986 − 0.164i)15-s − 0.250·16-s + (0.521 − 0.521i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.378 + 0.925i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.378 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.378 + 0.925i$
Analytic conductor: \(1.67685\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1/2),\ 0.378 + 0.925i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.833466 - 0.559591i\)
\(L(\frac12)\) \(\approx\) \(0.833466 - 0.559591i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
3 \( 1 + (-1.27 + 1.17i)T \)
5 \( 1 + (1.37 + 1.76i)T \)
7 \( 1 + (0.707 + 0.707i)T \)
good11 \( 1 + 2.48iT - 11T^{2} \)
13 \( 1 + (1.31 - 1.31i)T - 13iT^{2} \)
17 \( 1 + (-2.15 + 2.15i)T - 17iT^{2} \)
19 \( 1 + 1.26iT - 19T^{2} \)
23 \( 1 + (-3.83 - 3.83i)T + 23iT^{2} \)
29 \( 1 - 2.68T + 29T^{2} \)
31 \( 1 - 10.1T + 31T^{2} \)
37 \( 1 + (-3.73 - 3.73i)T + 37iT^{2} \)
41 \( 1 + 9.92iT - 41T^{2} \)
43 \( 1 + (7.18 - 7.18i)T - 43iT^{2} \)
47 \( 1 + (9.47 - 9.47i)T - 47iT^{2} \)
53 \( 1 + (-6.59 - 6.59i)T + 53iT^{2} \)
59 \( 1 + 1.66T + 59T^{2} \)
61 \( 1 + 1.70T + 61T^{2} \)
67 \( 1 + (4.58 + 4.58i)T + 67iT^{2} \)
71 \( 1 - 2.61iT - 71T^{2} \)
73 \( 1 + (-8.49 + 8.49i)T - 73iT^{2} \)
79 \( 1 - 3.26iT - 79T^{2} \)
83 \( 1 + (1.42 + 1.42i)T + 83iT^{2} \)
89 \( 1 - 11.6T + 89T^{2} \)
97 \( 1 + (0.983 + 0.983i)T + 97iT^{2} \)
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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

−12.21918518530705281075335423598, −11.38052680816973372194367819928, −9.817477462912383591606240860271, −8.999851815578247039514501079968, −8.139557043245834872757618608395, −7.37544579238740187359297786505, −6.28587354341666996855409816856, −4.74761995833898161111068221639, −3.14824810878796305445025086750, −1.00809102719402969751972245783, 2.49035819914518134272934781093, 3.49587184510857598534807987591, 4.75049874848850738806644150725, 6.70060336325445612276695919825, 7.86404605696063687137723163256, 8.583388231038849711217273109711, 10.00098872787039401505744587406, 10.20839369747832568221957663276, 11.44239353746232855240347562315, 12.37461340378158403011395415954

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