Properties

Label 2-690-115.22-c1-0-9
Degree $2$
Conductor $690$
Sign $-0.269 - 0.963i$
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 + (1.52 + 1.63i)5-s + 1.00·6-s + (−2.68 + 2.68i)7-s + (−0.707 + 0.707i)8-s − 1.00i·9-s + (−0.0835 + 2.23i)10-s + 0.403i·11-s + (0.707 + 0.707i)12-s + (−2.12 + 2.12i)13-s − 3.80·14-s + (2.23 + 0.0835i)15-s − 1.00·16-s + (−4.95 + 4.95i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (0.408 − 0.408i)3-s + 0.500i·4-s + (0.680 + 0.733i)5-s + 0.408·6-s + (−1.01 + 1.01i)7-s + (−0.250 + 0.250i)8-s − 0.333i·9-s + (−0.0264 + 0.706i)10-s + 0.121i·11-s + (0.204 + 0.204i)12-s + (−0.589 + 0.589i)13-s − 1.01·14-s + (0.576 + 0.0215i)15-s − 0.250·16-s + (−1.20 + 1.20i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.269 - 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.269 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.269 - 0.963i$
Analytic conductor: \(5.50967\)
Root analytic conductor: \(2.34727\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (367, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1/2),\ -0.269 - 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.23968 + 1.63407i\)
\(L(\frac12)\) \(\approx\) \(1.23968 + 1.63407i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.52 - 1.63i)T \)
23 \( 1 + (-4.10 + 2.47i)T \)
good7 \( 1 + (2.68 - 2.68i)T - 7iT^{2} \)
11 \( 1 - 0.403iT - 11T^{2} \)
13 \( 1 + (2.12 - 2.12i)T - 13iT^{2} \)
17 \( 1 + (4.95 - 4.95i)T - 17iT^{2} \)
19 \( 1 - 5.20T + 19T^{2} \)
29 \( 1 + 9.42iT - 29T^{2} \)
31 \( 1 - 3.33T + 31T^{2} \)
37 \( 1 + (-2.17 + 2.17i)T - 37iT^{2} \)
41 \( 1 - 5.69T + 41T^{2} \)
43 \( 1 + (2.56 + 2.56i)T + 43iT^{2} \)
47 \( 1 + (-8.64 - 8.64i)T + 47iT^{2} \)
53 \( 1 + (1.47 + 1.47i)T + 53iT^{2} \)
59 \( 1 - 13.4iT - 59T^{2} \)
61 \( 1 + 7.51iT - 61T^{2} \)
67 \( 1 + (-4.60 + 4.60i)T - 67iT^{2} \)
71 \( 1 + 5.45T + 71T^{2} \)
73 \( 1 + (-1.06 + 1.06i)T - 73iT^{2} \)
79 \( 1 - 0.383T + 79T^{2} \)
83 \( 1 + (-12.3 - 12.3i)T + 83iT^{2} \)
89 \( 1 - 6.23T + 89T^{2} \)
97 \( 1 + (-8.25 + 8.25i)T - 97iT^{2} \)
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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

−10.71362414176927986860484125171, −9.496151055356660751373508253813, −9.176201827842004823222387204959, −7.969757254807531861095358006198, −6.93110655868314284832040890566, −6.33333812273616379183950628520, −5.65664139571746003059502853954, −4.22558033860142705519004204350, −2.89919996548339587793170822295, −2.27005391063223300377600909943, 0.880122554499842944463007115843, 2.63027175419490801974570555854, 3.49450492829670723583010745978, 4.71720920292025468928008104822, 5.32675422956865991962181092798, 6.64706746818513233444479276252, 7.46267225824088352164069286633, 8.929231179901447406426780072208, 9.486459619613299490067275903711, 10.13229684744656137957757337148

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