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Properties

Label 2-690-115.68-c1-0-22
Degree $2$
Conductor $690$
Sign $-0.868 - 0.495i$
Analytic cond. $5.50967$
Root an. cond. $2.34727$
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 + (−0.397 + 2.20i)5-s − 1.00·6-s + (−1.66 − 1.66i)7-s + (−0.707 − 0.707i)8-s + 1.00i·9-s + (1.27 + 1.83i)10-s + 3.67i·11-s + (−0.707 + 0.707i)12-s + (−4.53 − 4.53i)13-s − 2.35·14-s + (1.83 − 1.27i)15-s − 1.00·16-s + (−4.03 − 4.03i)17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (−0.408 − 0.408i)3-s − 0.500i·4-s + (−0.177 + 0.984i)5-s − 0.408·6-s + (−0.630 − 0.630i)7-s + (−0.250 − 0.250i)8-s + 0.333i·9-s + (0.403 + 0.580i)10-s + 1.10i·11-s + (−0.204 + 0.204i)12-s + (−1.25 − 1.25i)13-s − 0.630·14-s + (0.474 − 0.329i)15-s − 0.250·16-s + (−0.978 − 0.978i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.868 - 0.495i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.868 - 0.495i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.868 - 0.495i$
Analytic conductor: \(5.50967\)
Root analytic conductor: \(2.34727\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (643, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1/2),\ -0.868 - 0.495i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0581128 + 0.219181i\)
\(L(\frac12)\) \(\approx\) \(0.0581128 + 0.219181i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.397 - 2.20i)T \)
23 \( 1 + (4.68 - 1.01i)T \)
good7 \( 1 + (1.66 + 1.66i)T + 7iT^{2} \)
11 \( 1 - 3.67iT - 11T^{2} \)
13 \( 1 + (4.53 + 4.53i)T + 13iT^{2} \)
17 \( 1 + (4.03 + 4.03i)T + 17iT^{2} \)
19 \( 1 + 0.786T + 19T^{2} \)
29 \( 1 - 4.68iT - 29T^{2} \)
31 \( 1 + 3.96T + 31T^{2} \)
37 \( 1 + (-4.59 - 4.59i)T + 37iT^{2} \)
41 \( 1 - 6.32T + 41T^{2} \)
43 \( 1 + (1.71 - 1.71i)T - 43iT^{2} \)
47 \( 1 + (5.71 - 5.71i)T - 47iT^{2} \)
53 \( 1 + (-5.93 + 5.93i)T - 53iT^{2} \)
59 \( 1 + 13.2iT - 59T^{2} \)
61 \( 1 + 3.15iT - 61T^{2} \)
67 \( 1 + (3.28 + 3.28i)T + 67iT^{2} \)
71 \( 1 - 6.19T + 71T^{2} \)
73 \( 1 + (-5.62 - 5.62i)T + 73iT^{2} \)
79 \( 1 + 11.6T + 79T^{2} \)
83 \( 1 + (5.05 - 5.05i)T - 83iT^{2} \)
89 \( 1 + 16.1T + 89T^{2} \)
97 \( 1 + (-2.18 - 2.18i)T + 97iT^{2} \)
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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

−9.992161377027484599969061461592, −9.705860179856532399593359122155, −7.87507094488155401951842705688, −7.11189034267907249108331068116, −6.55931353009826488951537245936, −5.32756692845961875744094063412, −4.35606164022048956242816767363, −3.11855406061340289424088053491, −2.19457781498593529227370234936, −0.097249272662725685425052093511, 2.31521650687497789628465182622, 3.94168285504297930343011252078, 4.51917076892450723199449993906, 5.73604099041718374075797973072, 6.16863684079971365279581692371, 7.37761048026599844735039761627, 8.568849198015363655496105474868, 9.059812088440243502217604946927, 9.944922182406437482681668792702, 11.22029321230776553128876454781

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