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Properties

Label 2-845-65.4-c1-0-30
Degree $2$
Conductor $845$
Sign $0.818 - 0.574i$
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  + (0.607 − 1.05i)2-s + (1.13 + 0.655i)3-s + (0.262 + 0.455i)4-s + (0.311 + 2.21i)5-s + (1.37 − 0.796i)6-s + (1.45 + 2.51i)7-s + 3.06·8-s + (−0.640 − 1.10i)9-s + (2.51 + 1.01i)10-s + (−0.185 − 0.107i)11-s + 0.688i·12-s + 3.52·14-s + (−1.09 + 2.71i)15-s + (1.33 − 2.31i)16-s + (−5.56 + 3.21i)17-s − 1.55·18-s + ⋯
L(s)  = 1  + (0.429 − 0.743i)2-s + (0.655 + 0.378i)3-s + (0.131 + 0.227i)4-s + (0.139 + 0.990i)5-s + (0.562 − 0.324i)6-s + (0.548 + 0.950i)7-s + 1.08·8-s + (−0.213 − 0.369i)9-s + (0.796 + 0.321i)10-s + (−0.0559 − 0.0323i)11-s + 0.198i·12-s + 0.942·14-s + (−0.283 + 0.701i)15-s + (0.334 − 0.578i)16-s + (−1.35 + 0.779i)17-s − 0.366·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.818 - 0.574i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.818 - 0.574i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $0.818 - 0.574i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (654, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ 0.818 - 0.574i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.63957 + 0.833324i\)
\(L(\frac12)\) \(\approx\) \(2.63957 + 0.833324i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.311 - 2.21i)T \)
13 \( 1 \)
good2 \( 1 + (-0.607 + 1.05i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-1.13 - 0.655i)T + (1.5 + 2.59i)T^{2} \)
7 \( 1 + (-1.45 - 2.51i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.185 + 0.107i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (5.56 - 3.21i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.91 + 1.10i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.06 + 2.34i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.35 + 7.54i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 5.59iT - 31T^{2} \)
37 \( 1 + (-1.14 + 1.97i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.64 - 1.52i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.50 + 3.18i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 1.09T + 47T^{2} \)
53 \( 1 - 6.23iT - 53T^{2} \)
59 \( 1 + (-8.02 + 4.63i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.140 - 0.243i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.88 + 6.72i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-5.26 + 3.04i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 10.2T + 73T^{2} \)
79 \( 1 - 14.2T + 79T^{2} \)
83 \( 1 + 9.52T + 83T^{2} \)
89 \( 1 + (4.86 + 2.80i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-9.02 - 15.6i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.46345567352501309762573230070, −9.548521095592790908143981954194, −8.573495591951157974697659417209, −7.956175769265797745671474972335, −6.76794510017954734276635129728, −5.90961488641833442919931582287, −4.50749934635369072308393504913, −3.69001907714041233066577195075, −2.63625443274584448577786710557, −2.16932909325876718810700479512, 1.19451284276438749688684187128, 2.29664765204884823590316885037, 4.08411870707193011467427886965, 4.84734159450850063226429887839, 5.61258536602011263096546177863, 6.79616382764354775587590091160, 7.56702333748130419936494419826, 8.148536530669971871594512345844, 9.056424004448756017711151744388, 10.04278927459788409120324748182

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