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Properties

Label 2-833-17.2-c1-0-9
Degree $2$
Conductor $833$
Sign $0.890 + 0.454i$
Analytic cond. $6.65153$
Root an. cond. $2.57905$
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  + (−1.07 − 1.07i)2-s + (−0.882 + 0.365i)3-s + 0.306i·4-s + (−0.477 − 1.15i)5-s + (1.34 + 0.555i)6-s + (−1.81 + 1.81i)8-s + (−1.47 + 1.47i)9-s + (−0.724 + 1.75i)10-s + (0.606 + 0.251i)11-s + (−0.111 − 0.270i)12-s − 1.93i·13-s + (0.843 + 0.843i)15-s + 4.51·16-s + (−1.69 + 3.75i)17-s + 3.16·18-s + (−0.0120 − 0.0120i)19-s + ⋯
L(s)  = 1  + (−0.759 − 0.759i)2-s + (−0.509 + 0.211i)3-s + 0.153i·4-s + (−0.213 − 0.515i)5-s + (0.547 + 0.226i)6-s + (−0.643 + 0.643i)8-s + (−0.491 + 0.491i)9-s + (−0.229 + 0.553i)10-s + (0.182 + 0.0757i)11-s + (−0.0323 − 0.0780i)12-s − 0.535i·13-s + (0.217 + 0.217i)15-s + 1.12·16-s + (−0.412 + 0.911i)17-s + 0.746·18-s + (−0.00276 − 0.00276i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 833 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.890 + 0.454i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 833 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.890 + 0.454i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(833\)    =    \(7^{2} \cdot 17\)
Sign: $0.890 + 0.454i$
Analytic conductor: \(6.65153\)
Root analytic conductor: \(2.57905\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{833} (393, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 833,\ (\ :1/2),\ 0.890 + 0.454i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.629854 - 0.151256i\)
\(L(\frac12)\) \(\approx\) \(0.629854 - 0.151256i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
17 \( 1 + (1.69 - 3.75i)T \)
good2 \( 1 + (1.07 + 1.07i)T + 2iT^{2} \)
3 \( 1 + (0.882 - 0.365i)T + (2.12 - 2.12i)T^{2} \)
5 \( 1 + (0.477 + 1.15i)T + (-3.53 + 3.53i)T^{2} \)
11 \( 1 + (-0.606 - 0.251i)T + (7.77 + 7.77i)T^{2} \)
13 \( 1 + 1.93iT - 13T^{2} \)
19 \( 1 + (0.0120 + 0.0120i)T + 19iT^{2} \)
23 \( 1 + (-6.35 - 2.63i)T + (16.2 + 16.2i)T^{2} \)
29 \( 1 + (-1.77 - 4.28i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 + (8.47 - 3.51i)T + (21.9 - 21.9i)T^{2} \)
37 \( 1 + (-1.59 + 0.659i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (-0.868 + 2.09i)T + (-28.9 - 28.9i)T^{2} \)
43 \( 1 + (0.170 - 0.170i)T - 43iT^{2} \)
47 \( 1 + 0.361iT - 47T^{2} \)
53 \( 1 + (-8.57 - 8.57i)T + 53iT^{2} \)
59 \( 1 + (-8.30 + 8.30i)T - 59iT^{2} \)
61 \( 1 + (-3.73 + 9.00i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 - 0.610T + 67T^{2} \)
71 \( 1 + (-11.2 + 4.66i)T + (50.2 - 50.2i)T^{2} \)
73 \( 1 + (-1.24 - 3.01i)T + (-51.6 + 51.6i)T^{2} \)
79 \( 1 + (-3.96 - 1.64i)T + (55.8 + 55.8i)T^{2} \)
83 \( 1 + (-3.30 - 3.30i)T + 83iT^{2} \)
89 \( 1 - 10.9iT - 89T^{2} \)
97 \( 1 + (-2.88 - 6.97i)T + (-68.5 + 68.5i)T^{2} \)
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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

−10.43229044973429594919485048728, −9.285610612245363226821783672684, −8.712489793698109028552064127967, −7.960669626803059434302785142117, −6.66802485168876857264363072502, −5.51586709465930486634711884844, −4.97959672107235727765640292615, −3.52057697267999341493661297662, −2.24253282415843320563459864807, −0.892026944019882208805834218707, 0.61893912352208977202956815927, 2.74454764177270866151587326696, 3.83487244151182464136906148393, 5.25282734901023398662083709285, 6.29788835085176480674373389574, 6.93781510580912002464967372908, 7.47122324967988151060930197811, 8.745894869145239062897535412510, 9.086741363829425691634793629934, 10.09931678328612888515441947772

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