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Properties

Label 2-690-115.103-c1-0-19
Degree $2$
Conductor $690$
Sign $-0.279 + 0.960i$
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  + (0.212 − 0.977i)2-s + (−0.800 + 0.599i)3-s + (−0.909 − 0.415i)4-s + (2.23 − 0.0484i)5-s + (0.415 + 0.909i)6-s + (0.261 − 3.65i)7-s + (−0.599 + 0.800i)8-s + (0.281 − 0.959i)9-s + (0.427 − 2.19i)10-s + (−0.378 + 0.588i)11-s + (0.977 − 0.212i)12-s + (0.0644 − 0.00460i)13-s + (−3.51 − 1.03i)14-s + (−1.76 + 1.37i)15-s + (0.654 + 0.755i)16-s + (0.623 − 1.67i)17-s + ⋯
L(s)  = 1  + (0.150 − 0.690i)2-s + (−0.462 + 0.345i)3-s + (−0.454 − 0.207i)4-s + (0.999 − 0.0216i)5-s + (0.169 + 0.371i)6-s + (0.0988 − 1.38i)7-s + (−0.211 + 0.283i)8-s + (0.0939 − 0.319i)9-s + (0.135 − 0.694i)10-s + (−0.114 + 0.177i)11-s + (0.282 − 0.0613i)12-s + (0.0178 − 0.00127i)13-s + (−0.939 − 0.276i)14-s + (−0.454 + 0.355i)15-s + (0.163 + 0.188i)16-s + (0.151 − 0.405i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.279 + 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.279 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.279 + 0.960i$
Analytic conductor: \(5.50967\)
Root analytic conductor: \(2.34727\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1/2),\ -0.279 + 0.960i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.877000 - 1.16814i\)
\(L(\frac12)\) \(\approx\) \(0.877000 - 1.16814i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.212 + 0.977i)T \)
3 \( 1 + (0.800 - 0.599i)T \)
5 \( 1 + (-2.23 + 0.0484i)T \)
23 \( 1 + (4.43 + 1.81i)T \)
good7 \( 1 + (-0.261 + 3.65i)T + (-6.92 - 0.996i)T^{2} \)
11 \( 1 + (0.378 - 0.588i)T + (-4.56 - 10.0i)T^{2} \)
13 \( 1 + (-0.0644 + 0.00460i)T + (12.8 - 1.85i)T^{2} \)
17 \( 1 + (-0.623 + 1.67i)T + (-12.8 - 11.1i)T^{2} \)
19 \( 1 + (-0.527 + 1.15i)T + (-12.4 - 14.3i)T^{2} \)
29 \( 1 + (-4.44 + 2.02i)T + (18.9 - 21.9i)T^{2} \)
31 \( 1 + (0.205 + 1.42i)T + (-29.7 + 8.73i)T^{2} \)
37 \( 1 + (4.20 + 7.70i)T + (-20.0 + 31.1i)T^{2} \)
41 \( 1 + (0.168 - 0.0494i)T + (34.4 - 22.1i)T^{2} \)
43 \( 1 + (1.91 + 2.56i)T + (-12.1 + 41.2i)T^{2} \)
47 \( 1 + (-3.36 + 3.36i)T - 47iT^{2} \)
53 \( 1 + (-11.7 - 0.842i)T + (52.4 + 7.54i)T^{2} \)
59 \( 1 + (2.90 + 2.51i)T + (8.39 + 58.3i)T^{2} \)
61 \( 1 + (6.39 - 0.919i)T + (58.5 - 17.1i)T^{2} \)
67 \( 1 + (2.90 + 0.631i)T + (60.9 + 27.8i)T^{2} \)
71 \( 1 + (-2.74 + 1.76i)T + (29.4 - 64.5i)T^{2} \)
73 \( 1 + (-1.15 + 0.429i)T + (55.1 - 47.8i)T^{2} \)
79 \( 1 + (10.9 - 12.6i)T + (-11.2 - 78.1i)T^{2} \)
83 \( 1 + (-12.6 + 6.89i)T + (44.8 - 69.8i)T^{2} \)
89 \( 1 + (2.23 - 15.5i)T + (-85.3 - 25.0i)T^{2} \)
97 \( 1 + (-0.329 - 0.179i)T + (52.4 + 81.6i)T^{2} \)
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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

−10.35331610742043230196673761986, −9.735463216565538908245582652982, −8.797848487617907833737430725301, −7.50156770305061714140299357412, −6.54353673080419171456110348791, −5.51487516206424774645842836738, −4.60512371655190690731068940024, −3.70490354611502667737213846401, −2.26387579280442797488173559055, −0.814712059976394071578048656979, 1.71308783755261394706216887107, 2.99186447084258586972963746426, 4.72287335828606672401876294801, 5.67935159678610593156747145818, 6.01625634059067740418023033608, 6.98280785151666183372236253831, 8.200569020708084801213548427505, 8.828318535761201985599499809061, 9.784638820151950803723591601770, 10.58308988038949906727431839261

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