Properties

Label 2-690-23.4-c1-0-0
Degree $2$
Conductor $690$
Sign $-0.927 - 0.373i$
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  + (−0.415 − 0.909i)2-s + (−0.959 + 0.281i)3-s + (−0.654 + 0.755i)4-s + (−0.841 − 0.540i)5-s + (0.654 + 0.755i)6-s + (0.626 + 4.35i)7-s + (0.959 + 0.281i)8-s + (0.841 − 0.540i)9-s + (−0.142 + 0.989i)10-s + (−0.307 + 0.673i)11-s + (0.415 − 0.909i)12-s + (0.494 − 3.43i)13-s + (3.70 − 2.38i)14-s + (0.959 + 0.281i)15-s + (−0.142 − 0.989i)16-s + (−5.19 − 5.99i)17-s + ⋯
L(s)  = 1  + (−0.293 − 0.643i)2-s + (−0.553 + 0.162i)3-s + (−0.327 + 0.377i)4-s + (−0.376 − 0.241i)5-s + (0.267 + 0.308i)6-s + (0.236 + 1.64i)7-s + (0.339 + 0.0996i)8-s + (0.280 − 0.180i)9-s + (−0.0450 + 0.313i)10-s + (−0.0927 + 0.203i)11-s + (0.119 − 0.262i)12-s + (0.137 − 0.953i)13-s + (0.990 − 0.636i)14-s + (0.247 + 0.0727i)15-s + (−0.0355 − 0.247i)16-s + (−1.25 − 1.45i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.927 - 0.373i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.927 - 0.373i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.927 - 0.373i$
Analytic conductor: \(5.50967\)
Root analytic conductor: \(2.34727\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1/2),\ -0.927 - 0.373i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00857499 + 0.0441995i\)
\(L(\frac12)\) \(\approx\) \(0.00857499 + 0.0441995i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.415 + 0.909i)T \)
3 \( 1 + (0.959 - 0.281i)T \)
5 \( 1 + (0.841 + 0.540i)T \)
23 \( 1 + (3.39 - 3.38i)T \)
good7 \( 1 + (-0.626 - 4.35i)T + (-6.71 + 1.97i)T^{2} \)
11 \( 1 + (0.307 - 0.673i)T + (-7.20 - 8.31i)T^{2} \)
13 \( 1 + (-0.494 + 3.43i)T + (-12.4 - 3.66i)T^{2} \)
17 \( 1 + (5.19 + 5.99i)T + (-2.41 + 16.8i)T^{2} \)
19 \( 1 + (1.66 - 1.92i)T + (-2.70 - 18.8i)T^{2} \)
29 \( 1 + (5.82 + 6.71i)T + (-4.12 + 28.7i)T^{2} \)
31 \( 1 + (-0.754 - 0.221i)T + (26.0 + 16.7i)T^{2} \)
37 \( 1 + (8.83 - 5.67i)T + (15.3 - 33.6i)T^{2} \)
41 \( 1 + (-9.19 - 5.90i)T + (17.0 + 37.2i)T^{2} \)
43 \( 1 + (0.408 - 0.120i)T + (36.1 - 23.2i)T^{2} \)
47 \( 1 + 6.92T + 47T^{2} \)
53 \( 1 + (-0.588 - 4.09i)T + (-50.8 + 14.9i)T^{2} \)
59 \( 1 + (-1.23 + 8.61i)T + (-56.6 - 16.6i)T^{2} \)
61 \( 1 + (3.04 + 0.894i)T + (51.3 + 32.9i)T^{2} \)
67 \( 1 + (5.32 + 11.6i)T + (-43.8 + 50.6i)T^{2} \)
71 \( 1 + (3.99 + 8.74i)T + (-46.4 + 53.6i)T^{2} \)
73 \( 1 + (2.74 - 3.16i)T + (-10.3 - 72.2i)T^{2} \)
79 \( 1 + (-0.0190 + 0.132i)T + (-75.7 - 22.2i)T^{2} \)
83 \( 1 + (10.4 - 6.69i)T + (34.4 - 75.4i)T^{2} \)
89 \( 1 + (7.81 - 2.29i)T + (74.8 - 48.1i)T^{2} \)
97 \( 1 + (8.12 + 5.21i)T + (40.2 + 88.2i)T^{2} \)
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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

−11.13346151648996614897946323453, −9.903298849467210645173827527162, −9.266852666551696328061130557895, −8.415827336121585698887279026439, −7.61416060400972260249907033477, −6.21442535397086271310008444016, −5.32604096403042442115872024590, −4.49580860252822450660532959735, −3.08308965408461517752556087929, −1.96451811411991632060170017239, 0.02746044117016939803936361084, 1.69346224050113056798985276801, 3.97882085248193928610320544211, 4.38473189116547551950584835134, 5.81030898023323978915971613452, 6.88757060632871660851828815436, 7.11558158069932458880733029821, 8.257552289994087477370960656195, 9.043203930926152261985486512860, 10.46524319843393637244491723633

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