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Properties

Label 2-845-13.9-c1-0-31
Degree $2$
Conductor $845$
Sign $0.945 + 0.326i$
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  + (0.109 − 0.190i)2-s + (0.800 − 1.38i)3-s + (0.975 + 1.69i)4-s − 5-s + (−0.175 − 0.304i)6-s + (0.166 + 0.287i)7-s + 0.868·8-s + (0.219 + 0.380i)9-s + (−0.109 + 0.190i)10-s + (2.68 − 4.65i)11-s + 3.12·12-s + 0.0729·14-s + (−0.800 + 1.38i)15-s + (−1.85 + 3.21i)16-s + (2.53 + 4.38i)17-s + 0.0965·18-s + ⋯
L(s)  = 1  + (0.0776 − 0.134i)2-s + (0.461 − 0.800i)3-s + (0.487 + 0.845i)4-s − 0.447·5-s + (−0.0717 − 0.124i)6-s + (0.0627 + 0.108i)7-s + 0.306·8-s + (0.0732 + 0.126i)9-s + (−0.0347 + 0.0601i)10-s + (0.809 − 1.40i)11-s + 0.901·12-s + 0.0195·14-s + (−0.206 + 0.357i)15-s + (−0.464 + 0.803i)16-s + (0.614 + 1.06i)17-s + 0.0227·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.945 + 0.326i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.945 + 0.326i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $0.945 + 0.326i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ 0.945 + 0.326i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.13937 - 0.359111i\)
\(L(\frac12)\) \(\approx\) \(2.13937 - 0.359111i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + T \)
13 \( 1 \)
good2 \( 1 + (-0.109 + 0.190i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-0.800 + 1.38i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + (-0.166 - 0.287i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.68 + 4.65i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-2.53 - 4.38i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.13 + 1.96i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.41 + 2.45i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.45 + 2.51i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 5.46T + 31T^{2} \)
37 \( 1 + (2.98 - 5.17i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.86 - 3.23i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.53 - 4.38i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 8.34T + 47T^{2} \)
53 \( 1 + 1.56T + 53T^{2} \)
59 \( 1 + (1.35 + 2.34i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (7.05 + 12.2i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.16 - 8.94i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (6.39 + 11.0i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 9.68T + 73T^{2} \)
79 \( 1 - 4.51T + 79T^{2} \)
83 \( 1 + 4.26T + 83T^{2} \)
89 \( 1 + (1.61 - 2.79i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.25 - 2.17i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.38565792632144662777846247839, −8.861357094963803282894111425957, −8.310557742799712034053596653713, −7.77683937823571588238638551536, −6.76475004828698614177311557403, −6.13597036965410603151596989588, −4.51811199762342635304867762504, −3.47993010247184810910762839184, −2.65555336664857331726064660184, −1.30971004310422116932270255964, 1.30944936176065371570760513281, 2.77679127338496074031461725912, 4.05525326636228590845451025980, 4.70809868960738500153116589279, 5.78704961679742922845219011369, 7.02110473565320575858576733155, 7.37101671307617330366116769888, 8.852216375159865870542367806220, 9.492549024044725666676270681664, 10.15667314526368206653265698929

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