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Properties

Label 2-690-15.8-c1-0-43
Degree $2$
Conductor $690$
Sign $-0.481 - 0.876i$
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 + (−0.292 − 1.70i)3-s + 1.00i·4-s + (−1.73 − 1.41i)5-s + (−0.999 + 1.41i)6-s + (2.44 − 2.44i)7-s + (0.707 − 0.707i)8-s + (−2.82 + i)9-s + (0.224 + 2.22i)10-s + 5.65i·11-s + (1.70 − 0.292i)12-s + (−3.44 − 3.44i)13-s − 3.46·14-s + (−1.90 + 3.37i)15-s − 1.00·16-s + (−5.19 − 5.19i)17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (−0.169 − 0.985i)3-s + 0.500i·4-s + (−0.774 − 0.632i)5-s + (−0.408 + 0.577i)6-s + (0.925 − 0.925i)7-s + (0.250 − 0.250i)8-s + (−0.942 + 0.333i)9-s + (0.0710 + 0.703i)10-s + 1.70i·11-s + (0.492 − 0.0845i)12-s + (−0.956 − 0.956i)13-s − 0.925·14-s + (−0.492 + 0.870i)15-s − 0.250·16-s + (−1.26 − 1.26i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.481 - 0.876i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.481 - 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.481 - 0.876i$
Analytic conductor: \(5.50967\)
Root analytic conductor: \(2.34727\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1/2),\ -0.481 - 0.876i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.189359 + 0.320104i\)
\(L(\frac12)\) \(\approx\) \(0.189359 + 0.320104i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.292 + 1.70i)T \)
5 \( 1 + (1.73 + 1.41i)T \)
23 \( 1 + (0.707 - 0.707i)T \)
good7 \( 1 + (-2.44 + 2.44i)T - 7iT^{2} \)
11 \( 1 - 5.65iT - 11T^{2} \)
13 \( 1 + (3.44 + 3.44i)T + 13iT^{2} \)
17 \( 1 + (5.19 + 5.19i)T + 17iT^{2} \)
19 \( 1 + 2.44iT - 19T^{2} \)
29 \( 1 + 2.19T + 29T^{2} \)
31 \( 1 - 4.89T + 31T^{2} \)
37 \( 1 + (2.44 - 2.44i)T - 37iT^{2} \)
41 \( 1 - 8.48iT - 41T^{2} \)
43 \( 1 + (-2.89 - 2.89i)T + 43iT^{2} \)
47 \( 1 + (4.87 + 4.87i)T + 47iT^{2} \)
53 \( 1 + (0.953 - 0.953i)T - 53iT^{2} \)
59 \( 1 + 5.51T + 59T^{2} \)
61 \( 1 - 0.898T + 61T^{2} \)
67 \( 1 + (10.8 - 10.8i)T - 67iT^{2} \)
71 \( 1 + 14.6iT - 71T^{2} \)
73 \( 1 + (1.89 + 1.89i)T + 73iT^{2} \)
79 \( 1 + 4.44iT - 79T^{2} \)
83 \( 1 + (-1.41 + 1.41i)T - 83iT^{2} \)
89 \( 1 - 9.12T + 89T^{2} \)
97 \( 1 + (0.449 - 0.449i)T - 97iT^{2} \)
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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

−9.926758225356391537147539913310, −8.961825851487191599791038705005, −7.87192451620844989124488123802, −7.52132725878053688166631733614, −6.85449573124089716066508959636, −4.86827672649006577011157868415, −4.58654977757880897072969566590, −2.75950670067094563259586136731, −1.54938482530216525334286388301, −0.23432258351520421328157749539, 2.33255341142453757616483201392, 3.74620680755994955245182727496, 4.70065284424148711728586149022, 5.78536824686983855798531591555, 6.49670991089733813716757518924, 7.84344209352500977275264726012, 8.608017746984505399175904609656, 9.004166572343823405240774961273, 10.30003235754950302492720695105, 11.00856697060805177698773486848

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