Properties

Label 2-690-115.114-c2-0-38
Degree $2$
Conductor $690$
Sign $-0.433 + 0.901i$
Analytic cond. $18.8011$
Root an. cond. $4.33602$
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  + 1.41i·2-s − 1.73i·3-s − 2.00·4-s + (3.16 − 3.86i)5-s + 2.44·6-s − 4.38·7-s − 2.82i·8-s − 2.99·9-s + (5.47 + 4.47i)10-s + 2.89i·11-s + 3.46i·12-s + 1.58i·13-s − 6.20i·14-s + (−6.70 − 5.48i)15-s + 4.00·16-s + 3.92·17-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.577i·3-s − 0.500·4-s + (0.633 − 0.773i)5-s + 0.408·6-s − 0.627·7-s − 0.353i·8-s − 0.333·9-s + (0.547 + 0.447i)10-s + 0.263i·11-s + 0.288i·12-s + 0.121i·13-s − 0.443i·14-s + (−0.446 − 0.365i)15-s + 0.250·16-s + 0.230·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.433 + 0.901i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.433 + 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.433 + 0.901i$
Analytic conductor: \(18.8011\)
Root analytic conductor: \(4.33602\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1),\ -0.433 + 0.901i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.9750699098\)
\(L(\frac12)\) \(\approx\) \(0.9750699098\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41iT \)
3 \( 1 + 1.73iT \)
5 \( 1 + (-3.16 + 3.86i)T \)
23 \( 1 + (-22.3 + 5.42i)T \)
good7 \( 1 + 4.38T + 49T^{2} \)
11 \( 1 - 2.89iT - 121T^{2} \)
13 \( 1 - 1.58iT - 169T^{2} \)
17 \( 1 - 3.92T + 289T^{2} \)
19 \( 1 + 23.3iT - 361T^{2} \)
29 \( 1 + 8.38T + 841T^{2} \)
31 \( 1 + 32.5T + 961T^{2} \)
37 \( 1 + 49.8T + 1.36e3T^{2} \)
41 \( 1 + 67.7T + 1.68e3T^{2} \)
43 \( 1 + 16.2T + 1.84e3T^{2} \)
47 \( 1 + 84.9iT - 2.20e3T^{2} \)
53 \( 1 - 1.17T + 2.80e3T^{2} \)
59 \( 1 + 103.T + 3.48e3T^{2} \)
61 \( 1 - 12.6iT - 3.72e3T^{2} \)
67 \( 1 - 125.T + 4.48e3T^{2} \)
71 \( 1 + 27.4T + 5.04e3T^{2} \)
73 \( 1 + 25.1iT - 5.32e3T^{2} \)
79 \( 1 + 25.4iT - 6.24e3T^{2} \)
83 \( 1 - 110.T + 6.88e3T^{2} \)
89 \( 1 + 14.4iT - 7.92e3T^{2} \)
97 \( 1 + 63.9T + 9.40e3T^{2} \)
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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

−9.678920266073313335171029782488, −9.013655694020377031151714124400, −8.328604881666283519205474655689, −7.08502639898387895991733091886, −6.62926922082927819523442415660, −5.48463766053485657355069372230, −4.84052308538587433391612883801, −3.33823048550450562939342290660, −1.82937534726069777745071846610, −0.33634876132175831718413851331, 1.70371086200310711350386899492, 3.10759996308265682029764571176, 3.60238335515389503854271862542, 5.09280503075387204045862858821, 5.92747547768346033668090289792, 6.92479402309821593151371397899, 8.128328819920553757090439444135, 9.254983440550296250604060867969, 9.759941409778958397070784430182, 10.57961224644620137502804851388

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