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Properties

Label 2-845-13.12-c1-0-34
Degree $2$
Conductor $845$
Sign $0.969 - 0.246i$
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  + 0.0240i·2-s + 2.93·3-s + 1.99·4-s + i·5-s + 0.0704i·6-s + 1.66i·7-s + 0.0960i·8-s + 5.58·9-s − 0.0240·10-s − 3.33i·11-s + 5.85·12-s − 0.0400·14-s + 2.93i·15-s + 3.99·16-s − 7.07·17-s + 0.134i·18-s + ⋯
L(s)  = 1  + 0.0169i·2-s + 1.69·3-s + 0.999·4-s + 0.447i·5-s + 0.0287i·6-s + 0.629i·7-s + 0.0339i·8-s + 1.86·9-s − 0.00759·10-s − 1.00i·11-s + 1.69·12-s − 0.0106·14-s + 0.756i·15-s + 0.999·16-s − 1.71·17-s + 0.0316i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 - 0.246i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.969 - 0.246i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $0.969 - 0.246i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (506, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ 0.969 - 0.246i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.27350 + 0.410408i\)
\(L(\frac12)\) \(\approx\) \(3.27350 + 0.410408i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - iT \)
13 \( 1 \)
good2 \( 1 - 0.0240iT - 2T^{2} \)
3 \( 1 - 2.93T + 3T^{2} \)
7 \( 1 - 1.66iT - 7T^{2} \)
11 \( 1 + 3.33iT - 11T^{2} \)
17 \( 1 + 7.07T + 17T^{2} \)
19 \( 1 + 6.68iT - 19T^{2} \)
23 \( 1 + 4.02T + 23T^{2} \)
29 \( 1 + 0.000401T + 29T^{2} \)
31 \( 1 - 4.14iT - 31T^{2} \)
37 \( 1 - 8.96iT - 37T^{2} \)
41 \( 1 - 9.60iT - 41T^{2} \)
43 \( 1 + 2.78T + 43T^{2} \)
47 \( 1 + 0.958iT - 47T^{2} \)
53 \( 1 + 3.68T + 53T^{2} \)
59 \( 1 + 7.99iT - 59T^{2} \)
61 \( 1 + 9.14T + 61T^{2} \)
67 \( 1 + 6.72iT - 67T^{2} \)
71 \( 1 + 4.38iT - 71T^{2} \)
73 \( 1 - 1.97iT - 73T^{2} \)
79 \( 1 + 3.77T + 79T^{2} \)
83 \( 1 - 3.64iT - 83T^{2} \)
89 \( 1 - 0.989iT - 89T^{2} \)
97 \( 1 + 18.4iT - 97T^{2} \)
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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

−10.13817152139515835674564950506, −9.144006431319685532282369783245, −8.548908161580919491947289126306, −7.83545673878168347384423225076, −6.82074703286345110513052319109, −6.25223586370776930585939045247, −4.67269783623339469262151544027, −3.28404735847132998010699170056, −2.75212181945934557118505874223, −1.89255753316345585249668830437, 1.78559658907062708249585339687, 2.33565902381253114532755148197, 3.73310792577118793267003834098, 4.31766554478492995173421719559, 5.96090974185994689364779028853, 7.14625883892990160442204716202, 7.56787616368285632538984144467, 8.415780461057627746164844203814, 9.251548194183566658651318490582, 10.07992974098795214049255469122

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