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Properties

Label 2-845-65.9-c1-0-16
Degree $2$
Conductor $845$
Sign $0.789 + 0.613i$
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  + (−2.20 − 1.27i)2-s + (−1.86 − 1.07i)3-s + (2.24 + 3.88i)4-s + (0.817 + 2.08i)5-s + (2.74 + 4.74i)6-s + (2.54 − 1.46i)7-s − 6.31i·8-s + (0.817 + 1.41i)9-s + (0.846 − 5.62i)10-s + (−0.317 + 0.550i)11-s − 9.64i·12-s − 7.48·14-s + (0.716 − 4.76i)15-s + (−3.55 + 6.16i)16-s + (−1.05 + 0.611i)17-s − 4.16i·18-s + ⋯
L(s)  = 1  + (−1.55 − 0.900i)2-s + (−1.07 − 0.621i)3-s + (1.12 + 1.94i)4-s + (0.365 + 0.930i)5-s + (1.11 + 1.93i)6-s + (0.961 − 0.555i)7-s − 2.23i·8-s + (0.272 + 0.472i)9-s + (0.267 − 1.78i)10-s + (−0.0957 + 0.165i)11-s − 2.78i·12-s − 1.99·14-s + (0.184 − 1.22i)15-s + (−0.889 + 1.54i)16-s + (−0.257 + 0.148i)17-s − 0.981i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.789 + 0.613i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.789 + 0.613i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $0.789 + 0.613i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (529, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ 0.789 + 0.613i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.501723 - 0.171874i\)
\(L(\frac12)\) \(\approx\) \(0.501723 - 0.171874i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.817 - 2.08i)T \)
13 \( 1 \)
good2 \( 1 + (2.20 + 1.27i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (1.86 + 1.07i)T + (1.5 + 2.59i)T^{2} \)
7 \( 1 + (-2.54 + 1.46i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.317 - 0.550i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.05 - 0.611i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.682 + 1.18i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.86 - 1.07i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.5 + 2.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 8.96T + 31T^{2} \)
37 \( 1 + (-1.05 - 0.611i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.98 - 8.62i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.18 + 0.683i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 6.16iT - 47T^{2} \)
53 \( 1 + 0.642iT - 53T^{2} \)
59 \( 1 + (3.79 + 6.57i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.13 - 1.96i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.95 - 4.01i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.31 - 2.28i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 10.3iT - 73T^{2} \)
79 \( 1 + 1.03T + 79T^{2} \)
83 \( 1 - 11.8iT - 83T^{2} \)
89 \( 1 + (-6.27 + 10.8i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-12.8 + 7.39i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.26503551224781892683753767703, −9.570321466711477463779235597802, −8.379516892027732772291795416311, −7.64866297924756222936236132080, −6.89789047644162910591755717101, −6.16610106425975668277865560583, −4.70148488726470232296235889223, −3.11233000498055923669887354490, −1.94755067426097771466190892402, −0.921689853934984183225469058625, 0.69589786392581122630811463078, 2.00987360332692649223861283851, 4.58735029903633389985500700551, 5.29934234562691772420726669448, 5.90844348971775568554857997771, 6.84300740846967170875641487552, 8.111210773597966629147838183938, 8.527959570304280302005419499938, 9.328076913892419805666884468724, 10.18434380840909874165243329031

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