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Properties

Label 2-833-17.8-c1-0-8
Degree $2$
Conductor $833$
Sign $-0.453 - 0.891i$
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  + (−1.37 + 1.37i)2-s + (−0.181 + 0.437i)3-s − 1.76i·4-s + (−3.14 − 1.30i)5-s + (−0.352 − 0.849i)6-s + (−0.317 − 0.317i)8-s + (1.96 + 1.96i)9-s + (6.10 − 2.52i)10-s + (0.314 + 0.760i)11-s + (0.774 + 0.320i)12-s − 0.168i·13-s + (1.14 − 1.14i)15-s + 4.40·16-s + (1.56 − 3.81i)17-s − 5.38·18-s + (2.05 − 2.05i)19-s + ⋯
L(s)  = 1  + (−0.970 + 0.970i)2-s + (−0.104 + 0.252i)3-s − 0.884i·4-s + (−1.40 − 0.582i)5-s + (−0.143 − 0.346i)6-s + (−0.112 − 0.112i)8-s + (0.654 + 0.654i)9-s + (1.93 − 0.799i)10-s + (0.0949 + 0.229i)11-s + (0.223 + 0.0926i)12-s − 0.0466i·13-s + (0.294 − 0.294i)15-s + 1.10·16-s + (0.380 − 0.924i)17-s − 1.27·18-s + (0.471 − 0.471i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 833 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.453 - 0.891i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 833 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.453 - 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(833\)    =    \(7^{2} \cdot 17\)
Sign: $-0.453 - 0.891i$
Analytic conductor: \(6.65153\)
Root analytic conductor: \(2.57905\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{833} (246, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 833,\ (\ :1/2),\ -0.453 - 0.891i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.296456 + 0.483771i\)
\(L(\frac12)\) \(\approx\) \(0.296456 + 0.483771i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.56 + 3.81i)T \)
good2 \( 1 + (1.37 - 1.37i)T - 2iT^{2} \)
3 \( 1 + (0.181 - 0.437i)T + (-2.12 - 2.12i)T^{2} \)
5 \( 1 + (3.14 + 1.30i)T + (3.53 + 3.53i)T^{2} \)
11 \( 1 + (-0.314 - 0.760i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 + 0.168iT - 13T^{2} \)
19 \( 1 + (-2.05 + 2.05i)T - 19iT^{2} \)
23 \( 1 + (2.39 + 5.79i)T + (-16.2 + 16.2i)T^{2} \)
29 \( 1 + (-6.61 - 2.74i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 + (1.08 - 2.62i)T + (-21.9 - 21.9i)T^{2} \)
37 \( 1 + (3.68 - 8.90i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (8.11 - 3.36i)T + (28.9 - 28.9i)T^{2} \)
43 \( 1 + (-4.65 - 4.65i)T + 43iT^{2} \)
47 \( 1 - 9.55iT - 47T^{2} \)
53 \( 1 + (2.45 - 2.45i)T - 53iT^{2} \)
59 \( 1 + (-0.449 - 0.449i)T + 59iT^{2} \)
61 \( 1 + (-6.44 + 2.66i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 - 4.45T + 67T^{2} \)
71 \( 1 + (1.62 - 3.92i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (-7.46 - 3.09i)T + (51.6 + 51.6i)T^{2} \)
79 \( 1 + (-2.89 - 6.97i)T + (-55.8 + 55.8i)T^{2} \)
83 \( 1 + (-10.6 + 10.6i)T - 83iT^{2} \)
89 \( 1 - 4.89iT - 89T^{2} \)
97 \( 1 + (3.70 + 1.53i)T + (68.5 + 68.5i)T^{2} \)
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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

−10.19340063742632823950795613622, −9.489486978499981862803504827907, −8.468667387157070680475346961886, −8.050045254597400755158590477139, −7.24890162605439435744785637288, −6.59357627611780166253548832110, −5.07398457941210378644133748616, −4.46981779890334900788148292548, −3.17209050195392459526598115414, −0.944582551469612437857243132271, 0.54926040127753123963468278180, 1.92007331710490909973101477886, 3.49798165072570111638507463504, 3.82761230514376228800921297635, 5.60946755797250258848070521128, 6.78175103439398449044237755952, 7.67115191905888897765645170822, 8.250415761164169343102014312165, 9.220950537050791401176891686571, 10.12832954027224586648177795709

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