Properties

Label 2-210-35.27-c1-0-6
Degree $2$
Conductor $210$
Sign $0.979 + 0.203i$
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 + (−0.707 − 0.707i)3-s + 1.00i·4-s + (1.19 − 1.88i)5-s − 1.00i·6-s + (0.510 − 2.59i)7-s + (−0.707 + 0.707i)8-s + 1.00i·9-s + (2.18 − 0.489i)10-s + 4.79·11-s + (0.707 − 0.707i)12-s + (0.585 + 0.585i)13-s + (2.19 − 1.47i)14-s + (−2.18 + 0.489i)15-s − 1.00·16-s + (−4.10 + 4.10i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (−0.408 − 0.408i)3-s + 0.500i·4-s + (0.535 − 0.844i)5-s − 0.408i·6-s + (0.192 − 0.981i)7-s + (−0.250 + 0.250i)8-s + 0.333i·9-s + (0.689 − 0.154i)10-s + 1.44·11-s + (0.204 − 0.204i)12-s + (0.162 + 0.162i)13-s + (0.587 − 0.394i)14-s + (−0.563 + 0.126i)15-s − 0.250·16-s + (−0.995 + 0.995i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 + 0.203i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.979 + 0.203i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.979 + 0.203i$
Analytic conductor: \(1.67685\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1/2),\ 0.979 + 0.203i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.50704 - 0.155221i\)
\(L(\frac12)\) \(\approx\) \(1.50704 - 0.155221i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 + (-1.19 + 1.88i)T \)
7 \( 1 + (-0.510 + 2.59i)T \)
good11 \( 1 - 4.79T + 11T^{2} \)
13 \( 1 + (-0.585 - 0.585i)T + 13iT^{2} \)
17 \( 1 + (4.10 - 4.10i)T - 17iT^{2} \)
19 \( 1 + 2.36T + 19T^{2} \)
23 \( 1 + (-2.97 + 2.97i)T - 23iT^{2} \)
29 \( 1 - 9.94iT - 29T^{2} \)
31 \( 1 + 3.02iT - 31T^{2} \)
37 \( 1 + (6.10 + 6.10i)T + 37iT^{2} \)
41 \( 1 - 10.9iT - 41T^{2} \)
43 \( 1 + (5.74 - 5.74i)T - 43iT^{2} \)
47 \( 1 + (-0.363 + 0.363i)T - 47iT^{2} \)
53 \( 1 + (-2.36 + 2.36i)T - 53iT^{2} \)
59 \( 1 + 2.07T + 59T^{2} \)
61 \( 1 - 5.55iT - 61T^{2} \)
67 \( 1 + (0.979 + 0.979i)T + 67iT^{2} \)
71 \( 1 - 5.25T + 71T^{2} \)
73 \( 1 + (-6.11 - 6.11i)T + 73iT^{2} \)
79 \( 1 - 5.10iT - 79T^{2} \)
83 \( 1 + (3.22 + 3.22i)T + 83iT^{2} \)
89 \( 1 - 11.8T + 89T^{2} \)
97 \( 1 + (-8.05 + 8.05i)T - 97iT^{2} \)
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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

−12.68783165805148074475871880864, −11.52707580121522739707516715821, −10.57263465879277589305692942192, −9.141667590251107116941742814791, −8.317504352647498040801339402788, −6.87001730275334379943825160374, −6.30468670130691745772430278441, −4.89091187026500027028296423469, −3.97739024724358775341280596756, −1.51127531557405934118958132210, 2.14154726512495463862268014475, 3.56317202800352766520120139633, 4.96664064393783025382385248973, 6.08357376596615012386997862876, 6.86657312359812776297186687532, 8.864470491037098176096885498928, 9.573660821346382307841723429040, 10.66881903654494686921305557068, 11.56043057792626226741196199905, 12.04198704094714541496910004333

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