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Properties

Label 2-833-17.13-c1-0-41
Degree $2$
Conductor $833$
Sign $-0.998 - 0.0591i$
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  − 1.94i·2-s + (−1.10 − 1.10i)3-s − 1.80·4-s + (1.50 + 1.50i)5-s + (−2.14 + 2.14i)6-s − 0.386i·8-s − 0.575i·9-s + (2.93 − 2.93i)10-s + (0.794 − 0.794i)11-s + (1.98 + 1.98i)12-s + 2.44·13-s − 3.31i·15-s − 4.35·16-s + (3.39 − 2.34i)17-s − 1.12·18-s + 0.402i·19-s + ⋯
L(s)  = 1  − 1.37i·2-s + (−0.635 − 0.635i)3-s − 0.900·4-s + (0.673 + 0.673i)5-s + (−0.876 + 0.876i)6-s − 0.136i·8-s − 0.191i·9-s + (0.928 − 0.928i)10-s + (0.239 − 0.239i)11-s + (0.572 + 0.572i)12-s + 0.679·13-s − 0.856i·15-s − 1.08·16-s + (0.823 − 0.567i)17-s − 0.264·18-s + 0.0923i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 833 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0591i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 833 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0591i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(833\)    =    \(7^{2} \cdot 17\)
Sign: $-0.998 - 0.0591i$
Analytic conductor: \(6.65153\)
Root analytic conductor: \(2.57905\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{833} (540, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 833,\ (\ :1/2),\ -0.998 - 0.0591i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0396320 + 1.33807i\)
\(L(\frac12)\) \(\approx\) \(0.0396320 + 1.33807i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.39 + 2.34i)T \)
good2 \( 1 + 1.94iT - 2T^{2} \)
3 \( 1 + (1.10 + 1.10i)T + 3iT^{2} \)
5 \( 1 + (-1.50 - 1.50i)T + 5iT^{2} \)
11 \( 1 + (-0.794 + 0.794i)T - 11iT^{2} \)
13 \( 1 - 2.44T + 13T^{2} \)
19 \( 1 - 0.402iT - 19T^{2} \)
23 \( 1 + (-3.18 + 3.18i)T - 23iT^{2} \)
29 \( 1 + (7.19 + 7.19i)T + 29iT^{2} \)
31 \( 1 + (0.346 + 0.346i)T + 31iT^{2} \)
37 \( 1 + (0.748 + 0.748i)T + 37iT^{2} \)
41 \( 1 + (1.11 - 1.11i)T - 41iT^{2} \)
43 \( 1 - 6.77iT - 43T^{2} \)
47 \( 1 + 8.25T + 47T^{2} \)
53 \( 1 + 7.68iT - 53T^{2} \)
59 \( 1 - 0.800iT - 59T^{2} \)
61 \( 1 + (-8.04 + 8.04i)T - 61iT^{2} \)
67 \( 1 + 6.85T + 67T^{2} \)
71 \( 1 + (-10.4 - 10.4i)T + 71iT^{2} \)
73 \( 1 + (-10.9 - 10.9i)T + 73iT^{2} \)
79 \( 1 + (-5.70 + 5.70i)T - 79iT^{2} \)
83 \( 1 - 6.72iT - 83T^{2} \)
89 \( 1 - 6.04T + 89T^{2} \)
97 \( 1 + (6.33 + 6.33i)T + 97iT^{2} \)
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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

−9.824801194760432859649327578713, −9.512492603686980982445151239217, −8.187325287738250027969175543815, −6.89485083055123679636416095934, −6.37321174296032596653675293157, −5.40673183628390580681802194994, −3.91655722311916178346395019581, −2.99719843728293992460933386974, −1.88889047429212242747777632451, −0.74418375167959742827759689723, 1.68280642151316335914492198812, 3.70540983427784848485272063273, 5.00208851135673391909041462480, 5.38238044371203488100192757233, 6.10090534768252301575931693385, 7.11985227729961923265760680750, 7.993792257331807521283020253241, 8.936985262830973351693073801838, 9.517469659318874904319116562859, 10.60950093616974423575318770997

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