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Properties

Label 2-690-115.22-c1-0-15
Degree $2$
Conductor $690$
Sign $0.515 + 0.857i$
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 + (0.899 + 2.04i)5-s − 1.00·6-s + (1.80 − 1.80i)7-s + (0.707 − 0.707i)8-s − 1.00i·9-s + (0.811 − 2.08i)10-s − 4.16i·11-s + (0.707 + 0.707i)12-s + (−3.18 + 3.18i)13-s − 2.55·14-s + (2.08 + 0.811i)15-s − 1.00·16-s + (1.68 − 1.68i)17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (0.408 − 0.408i)3-s + 0.500i·4-s + (0.402 + 0.915i)5-s − 0.408·6-s + (0.683 − 0.683i)7-s + (0.250 − 0.250i)8-s − 0.333i·9-s + (0.256 − 0.658i)10-s − 1.25i·11-s + (0.204 + 0.204i)12-s + (−0.884 + 0.884i)13-s − 0.683·14-s + (0.537 + 0.209i)15-s − 0.250·16-s + (0.408 − 0.408i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.515 + 0.857i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.515 + 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.515 + 0.857i$
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: \((2,\ 690,\ (\ :1/2),\ 0.515 + 0.857i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.33794 - 0.756944i\)
\(L(\frac12)\) \(\approx\) \(1.33794 - 0.756944i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.899 - 2.04i)T \)
23 \( 1 + (-1.98 + 4.36i)T \)
good7 \( 1 + (-1.80 + 1.80i)T - 7iT^{2} \)
11 \( 1 + 4.16iT - 11T^{2} \)
13 \( 1 + (3.18 - 3.18i)T - 13iT^{2} \)
17 \( 1 + (-1.68 + 1.68i)T - 17iT^{2} \)
19 \( 1 - 5.24T + 19T^{2} \)
29 \( 1 - 3.48iT - 29T^{2} \)
31 \( 1 - 7.27T + 31T^{2} \)
37 \( 1 + (-5.46 + 5.46i)T - 37iT^{2} \)
41 \( 1 - 4.53T + 41T^{2} \)
43 \( 1 + (-6.37 - 6.37i)T + 43iT^{2} \)
47 \( 1 + (9.09 + 9.09i)T + 47iT^{2} \)
53 \( 1 + (6.56 + 6.56i)T + 53iT^{2} \)
59 \( 1 - 6.80iT - 59T^{2} \)
61 \( 1 - 0.759iT - 61T^{2} \)
67 \( 1 + (-2.02 + 2.02i)T - 67iT^{2} \)
71 \( 1 + 9.56T + 71T^{2} \)
73 \( 1 + (5.78 - 5.78i)T - 73iT^{2} \)
79 \( 1 + 6.84T + 79T^{2} \)
83 \( 1 + (2.40 + 2.40i)T + 83iT^{2} \)
89 \( 1 + 3.58T + 89T^{2} \)
97 \( 1 + (-4.03 + 4.03i)T - 97iT^{2} \)
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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

−10.27485739693199589380443001511, −9.578336732551494506254138197737, −8.648151559739435373546952437630, −7.65038223209453355483696873078, −7.12795915371454473408437031610, −6.07036507240326630797425458937, −4.65072205195152387023380129256, −3.31526773919444845672430281603, −2.50198123369895939489906896590, −1.08804616767754430628883659663, 1.43261491911079647420824152420, 2.70297099357277537137724441134, 4.55851196807891018609354184294, 5.12349856966245822422258414254, 5.98536948581290830820564640565, 7.58223039472692296242792825191, 7.924092124083582946629392749248, 8.950423202091119943421975490156, 9.756216199794208561490109789668, 10.02198122398187413274364580954

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