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Properties

Label 2-845-169.114-c1-0-48
Degree $2$
Conductor $845$
Sign $-0.999 - 0.0150i$
Analytic cond. $6.74735$
Root an. cond. $2.59756$
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  + (−1.20 − 0.982i)2-s + (2.59 + 0.209i)3-s + (0.0822 + 0.402i)4-s + (−0.992 + 0.120i)5-s + (−2.91 − 2.79i)6-s + (−2.51 − 3.97i)7-s + (−1.14 + 2.18i)8-s + (3.72 + 0.605i)9-s + (1.31 + 0.829i)10-s + (0.517 + 3.18i)11-s + (0.128 + 1.06i)12-s + (−2.44 − 2.64i)13-s + (−0.879 + 7.24i)14-s + (−2.60 + 0.104i)15-s + (4.28 − 1.82i)16-s + (4.74 − 3.00i)17-s + ⋯
L(s)  = 1  + (−0.850 − 0.694i)2-s + (1.49 + 0.120i)3-s + (0.0411 + 0.201i)4-s + (−0.443 + 0.0539i)5-s + (−1.18 − 1.14i)6-s + (−0.949 − 1.50i)7-s + (−0.405 + 0.772i)8-s + (1.24 + 0.201i)9-s + (0.415 + 0.262i)10-s + (0.155 + 0.959i)11-s + (0.0372 + 0.306i)12-s + (−0.679 − 0.733i)13-s + (−0.235 + 1.93i)14-s + (−0.671 + 0.0270i)15-s + (1.07 − 0.455i)16-s + (1.15 − 0.728i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0150i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0150i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $-0.999 - 0.0150i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (621, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ -0.999 - 0.0150i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00582833 + 0.773057i\)
\(L(\frac12)\) \(\approx\) \(0.00582833 + 0.773057i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.992 - 0.120i)T \)
13 \( 1 + (2.44 + 2.64i)T \)
good2 \( 1 + (1.20 + 0.982i)T + (0.400 + 1.95i)T^{2} \)
3 \( 1 + (-2.59 - 0.209i)T + (2.96 + 0.481i)T^{2} \)
7 \( 1 + (2.51 + 3.97i)T + (-3.00 + 6.32i)T^{2} \)
11 \( 1 + (-0.517 - 3.18i)T + (-10.4 + 3.48i)T^{2} \)
17 \( 1 + (-4.74 + 3.00i)T + (7.28 - 15.3i)T^{2} \)
19 \( 1 + (5.09 + 2.93i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.26 + 3.92i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (5.51 - 6.75i)T + (-5.80 - 28.4i)T^{2} \)
31 \( 1 + (0.479 - 1.94i)T + (-27.4 - 14.4i)T^{2} \)
37 \( 1 + (6.71 - 1.94i)T + (31.2 - 19.7i)T^{2} \)
41 \( 1 + (-0.805 + 9.97i)T + (-40.4 - 6.57i)T^{2} \)
43 \( 1 + (-1.15 + 3.98i)T + (-36.3 - 22.9i)T^{2} \)
47 \( 1 + (2.25 - 2.54i)T + (-5.66 - 46.6i)T^{2} \)
53 \( 1 + (6.73 + 3.53i)T + (30.1 + 43.6i)T^{2} \)
59 \( 1 + (0.200 - 0.470i)T + (-40.8 - 42.5i)T^{2} \)
61 \( 1 + (-0.200 + 4.98i)T + (-60.8 - 4.90i)T^{2} \)
67 \( 1 + (-9.57 - 1.95i)T + (61.6 + 26.2i)T^{2} \)
71 \( 1 + (-10.6 + 5.04i)T + (44.9 - 54.9i)T^{2} \)
73 \( 1 + (-4.99 - 1.89i)T + (54.6 + 48.4i)T^{2} \)
79 \( 1 + (-3.92 - 3.47i)T + (9.52 + 78.4i)T^{2} \)
83 \( 1 + (-0.652 + 0.450i)T + (29.4 - 77.6i)T^{2} \)
89 \( 1 + (0.747 - 0.431i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.39 + 7.18i)T + (-26.9 - 93.1i)T^{2} \)
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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

−9.777994688083373904668573302785, −9.150553306116881834553881741215, −8.237209625609721458212139314138, −7.45663155056852145308291364854, −6.86493219379163195373779726507, −5.01468625550772835629424045475, −3.79548610964873425392080044727, −3.10011440192950608719445050205, −2.01385601456660057969414507818, −0.39069244875823613727949936067, 2.07100741757743993259187842639, 3.25833043192675413745241543205, 3.82736620800113192879258689432, 5.80505874320017674036135175178, 6.46305731587928553660217717066, 7.68926564808350490828417138985, 8.169722490280375966177010124002, 8.765847032841361160711620934938, 9.494570757382457457251008703897, 9.867713519855627005756262710198

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