Properties

Label 2-370-185.142-c1-0-13
Degree $2$
Conductor $370$
Sign $0.857 + 0.515i$
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 + (0.215 + 0.215i)3-s − 4-s + (0.336 − 2.21i)5-s + (−0.215 + 0.215i)6-s + (−0.673 − 0.673i)7-s i·8-s − 2.90i·9-s + (2.21 + 0.336i)10-s − 1.69i·11-s + (−0.215 − 0.215i)12-s − 2.06i·13-s + (0.673 − 0.673i)14-s + (0.548 − 0.403i)15-s + 16-s + 2.58·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.124 + 0.124i)3-s − 0.5·4-s + (0.150 − 0.988i)5-s + (−0.0878 + 0.0878i)6-s + (−0.254 − 0.254i)7-s − 0.353i·8-s − 0.969i·9-s + (0.699 + 0.106i)10-s − 0.512i·11-s + (−0.0621 − 0.0621i)12-s − 0.571i·13-s + (0.179 − 0.179i)14-s + (0.141 − 0.104i)15-s + 0.250·16-s + 0.625·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.857 + 0.515i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.857 + 0.515i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.857 + 0.515i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (327, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ 0.857 + 0.515i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.19546 - 0.331567i\)
\(L(\frac12)\) \(\approx\) \(1.19546 - 0.331567i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.336 + 2.21i)T \)
37 \( 1 + (-5.90 - 1.45i)T \)
good3 \( 1 + (-0.215 - 0.215i)T + 3iT^{2} \)
7 \( 1 + (0.673 + 0.673i)T + 7iT^{2} \)
11 \( 1 + 1.69iT - 11T^{2} \)
13 \( 1 + 2.06iT - 13T^{2} \)
17 \( 1 - 2.58T + 17T^{2} \)
19 \( 1 + (0.312 + 0.312i)T + 19iT^{2} \)
23 \( 1 - 3.40iT - 23T^{2} \)
29 \( 1 + (-2.85 + 2.85i)T - 29iT^{2} \)
31 \( 1 + (0.382 + 0.382i)T + 31iT^{2} \)
41 \( 1 - 3.11iT - 41T^{2} \)
43 \( 1 - 7.38iT - 43T^{2} \)
47 \( 1 + (-1.26 - 1.26i)T + 47iT^{2} \)
53 \( 1 + (2.69 - 2.69i)T - 53iT^{2} \)
59 \( 1 + (-3.61 - 3.61i)T + 59iT^{2} \)
61 \( 1 + (4.54 + 4.54i)T + 61iT^{2} \)
67 \( 1 + (-8.19 + 8.19i)T - 67iT^{2} \)
71 \( 1 + 8.12T + 71T^{2} \)
73 \( 1 + (-3.81 - 3.81i)T + 73iT^{2} \)
79 \( 1 + (3.33 + 3.33i)T + 79iT^{2} \)
83 \( 1 + (6.48 - 6.48i)T - 83iT^{2} \)
89 \( 1 + (0.486 - 0.486i)T - 89iT^{2} \)
97 \( 1 - 0.430T + 97T^{2} \)
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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

−11.44305516889663357489214132758, −9.993780616711061160459800004558, −9.426687083318991865757101783066, −8.471217760220782995410297645597, −7.70449693262569493758601870526, −6.37799414574385367022228959388, −5.61581064692847615914082863175, −4.44446496079738331055840489169, −3.30457411713041323180116144858, −0.880551271798739073842238232886, 1.99620226903074505290369470278, 2.95856124031334565764723489765, 4.30839935129956202510573597066, 5.57967772947146728396997492523, 6.80809926112869128609334019063, 7.73411140819758395409533974446, 8.854992802781847509474465848507, 9.937779997003719780903606826643, 10.53639025212512101655074353897, 11.37919022603785936344495108117

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