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Properties

Label 2-370-185.43-c1-0-5
Degree $2$
Conductor $370$
Sign $-0.997 - 0.0699i$
Analytic cond. $2.95446$
Root an. cond. $1.71885$
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  + i·2-s + (−1.28 + 1.28i)3-s − 4-s + (1.45 + 1.69i)5-s + (−1.28 − 1.28i)6-s + (0.579 − 0.579i)7-s − i·8-s − 0.324i·9-s + (−1.69 + 1.45i)10-s + 3.64i·11-s + (1.28 − 1.28i)12-s + 3.10i·13-s + (0.579 + 0.579i)14-s + (−4.06 − 0.301i)15-s + 16-s − 8.09·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.744 + 0.744i)3-s − 0.5·4-s + (0.652 + 0.757i)5-s + (−0.526 − 0.526i)6-s + (0.219 − 0.219i)7-s − 0.353i·8-s − 0.108i·9-s + (−0.535 + 0.461i)10-s + 1.09i·11-s + (0.372 − 0.372i)12-s + 0.861i·13-s + (0.154 + 0.154i)14-s + (−1.04 − 0.0777i)15-s + 0.250·16-s − 1.96·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0699i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0699i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $-0.997 - 0.0699i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ -0.997 - 0.0699i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0330162 + 0.942496i\)
\(L(\frac12)\) \(\approx\) \(0.0330162 + 0.942496i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
5 \( 1 + (-1.45 - 1.69i)T \)
37 \( 1 + (-3.63 - 4.87i)T \)
good3 \( 1 + (1.28 - 1.28i)T - 3iT^{2} \)
7 \( 1 + (-0.579 + 0.579i)T - 7iT^{2} \)
11 \( 1 - 3.64iT - 11T^{2} \)
13 \( 1 - 3.10iT - 13T^{2} \)
17 \( 1 + 8.09T + 17T^{2} \)
19 \( 1 + (-3.05 + 3.05i)T - 19iT^{2} \)
23 \( 1 + 7.79iT - 23T^{2} \)
29 \( 1 + (1.37 + 1.37i)T + 29iT^{2} \)
31 \( 1 + (-3.23 + 3.23i)T - 31iT^{2} \)
41 \( 1 - 9.31iT - 41T^{2} \)
43 \( 1 - 10.9iT - 43T^{2} \)
47 \( 1 + (4.11 - 4.11i)T - 47iT^{2} \)
53 \( 1 + (-0.446 - 0.446i)T + 53iT^{2} \)
59 \( 1 + (-6.16 + 6.16i)T - 59iT^{2} \)
61 \( 1 + (-8.71 + 8.71i)T - 61iT^{2} \)
67 \( 1 + (-2.01 - 2.01i)T + 67iT^{2} \)
71 \( 1 - 0.00151T + 71T^{2} \)
73 \( 1 + (-9.32 + 9.32i)T - 73iT^{2} \)
79 \( 1 + (0.760 - 0.760i)T - 79iT^{2} \)
83 \( 1 + (-3.16 - 3.16i)T + 83iT^{2} \)
89 \( 1 + (-5.80 - 5.80i)T + 89iT^{2} \)
97 \( 1 + 14.6T + 97T^{2} \)
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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

−11.41845066277581581445211825172, −10.99091944303629590743348984671, −9.852162492428735387638961191770, −9.408501363244654156689892485771, −7.990145191969927397518665600033, −6.70981618914162584940342364024, −6.36055283726904603025853389945, −4.78112270028893963176439011894, −4.47344278654808169290170836685, −2.36019406771387180579590747974, 0.70432887695903331489632602672, 2.02262626103259848706792541555, 3.70089143774324362716893420247, 5.36288670953957310087387848214, 5.74345618318177744399321811109, 7.05508921254849183773584715597, 8.427139723470004487249297316626, 9.074076561616839176519902406338, 10.19682599243993089709107087331, 11.20854829012298215960919993080

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