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Properties

Label 2-845-13.3-c1-0-46
Degree $2$
Conductor $845$
Sign $-0.872 + 0.488i$
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  + (−0.207 − 0.358i)2-s + (−0.707 − 1.22i)3-s + (0.914 − 1.58i)4-s + 5-s + (−0.292 + 0.507i)6-s + (0.414 − 0.717i)7-s − 1.58·8-s + (0.500 − 0.866i)9-s + (−0.207 − 0.358i)10-s + (−0.292 − 0.507i)11-s − 2.58·12-s − 0.343·14-s + (−0.707 − 1.22i)15-s + (−1.49 − 2.59i)16-s + (2.41 − 4.18i)17-s − 0.414·18-s + ⋯
L(s)  = 1  + (−0.146 − 0.253i)2-s + (−0.408 − 0.707i)3-s + (0.457 − 0.791i)4-s + 0.447·5-s + (−0.119 + 0.207i)6-s + (0.156 − 0.271i)7-s − 0.560·8-s + (0.166 − 0.288i)9-s + (−0.0654 − 0.113i)10-s + (−0.0883 − 0.152i)11-s − 0.746·12-s − 0.0917·14-s + (−0.182 − 0.316i)15-s + (−0.374 − 0.649i)16-s + (0.585 − 1.01i)17-s − 0.0976·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.872 + 0.488i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.872 + 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $-0.872 + 0.488i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (146, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ -0.872 + 0.488i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.352151 - 1.34878i\)
\(L(\frac12)\) \(\approx\) \(0.352151 - 1.34878i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.207 + 0.358i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (0.707 + 1.22i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (-0.414 + 0.717i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.292 + 0.507i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-2.41 + 4.18i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.70 - 2.95i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.707 - 1.22i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.82 + 4.89i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 10.2T + 31T^{2} \)
37 \( 1 + (4.24 + 7.34i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.41 - 7.64i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.53 - 2.65i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 0.828T + 47T^{2} \)
53 \( 1 + 14.4T + 53T^{2} \)
59 \( 1 + (5.12 - 8.87i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4 + 6.92i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1 - 1.73i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3.94 + 6.84i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 8.48T + 73T^{2} \)
79 \( 1 - 8.48T + 79T^{2} \)
83 \( 1 + 8.82T + 83T^{2} \)
89 \( 1 + (3 + 5.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.82 - 3.16i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.816146467145774472299353347086, −9.359753109309886328346574932510, −7.966985551874519800943807755014, −7.14787305973108074570996120835, −6.24052934961422595554113917239, −5.76510288896007166397074387518, −4.57522932446634176927990328914, −3.00162019023852644275567254205, −1.76557519005309632906145936520, −0.75417418157699704952913534633, 1.95889447291309011866925635794, 3.18040624542011556755735774814, 4.32846172225717505233816277063, 5.26096721842027611609768949040, 6.23473579132421504279672423367, 7.08066109336215834689609311092, 8.105775524522688958714623992922, 8.753395750509341909616757707796, 9.801767139713277550434422745017, 10.55161349929559418536308597411

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