Properties

Label 2-370-185.68-c1-0-2
Degree $2$
Conductor $370$
Sign $-0.993 - 0.117i$
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  − 2-s + (−1.28 + 1.28i)3-s + 4-s + (1.63 + 1.52i)5-s + (1.28 − 1.28i)6-s + (−3.01 + 3.01i)7-s − 8-s − 0.323i·9-s + (−1.63 − 1.52i)10-s − 2.38i·11-s + (−1.28 + 1.28i)12-s + 1.44·13-s + (3.01 − 3.01i)14-s + (−4.07 + 0.148i)15-s + 16-s + 2.09i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.744 + 0.744i)3-s + 0.5·4-s + (0.732 + 0.680i)5-s + (0.526 − 0.526i)6-s + (−1.13 + 1.13i)7-s − 0.353·8-s − 0.107i·9-s + (−0.517 − 0.481i)10-s − 0.720i·11-s + (−0.372 + 0.372i)12-s + 0.401·13-s + (0.805 − 0.805i)14-s + (−1.05 + 0.0384i)15-s + 0.250·16-s + 0.508i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 - 0.117i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.993 - 0.117i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $-0.993 - 0.117i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (253, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ -0.993 - 0.117i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0287543 + 0.487122i\)
\(L(\frac12)\) \(\approx\) \(0.0287543 + 0.487122i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
5 \( 1 + (-1.63 - 1.52i)T \)
37 \( 1 + (1.89 + 5.78i)T \)
good3 \( 1 + (1.28 - 1.28i)T - 3iT^{2} \)
7 \( 1 + (3.01 - 3.01i)T - 7iT^{2} \)
11 \( 1 + 2.38iT - 11T^{2} \)
13 \( 1 - 1.44T + 13T^{2} \)
17 \( 1 - 2.09iT - 17T^{2} \)
19 \( 1 + (2.46 + 2.46i)T + 19iT^{2} \)
23 \( 1 + 0.168T + 23T^{2} \)
29 \( 1 + (4.74 - 4.74i)T - 29iT^{2} \)
31 \( 1 + (3.34 + 3.34i)T + 31iT^{2} \)
41 \( 1 - 6.22iT - 41T^{2} \)
43 \( 1 + 3.07T + 43T^{2} \)
47 \( 1 + (4.67 - 4.67i)T - 47iT^{2} \)
53 \( 1 + (2.87 + 2.87i)T + 53iT^{2} \)
59 \( 1 + (-6.78 - 6.78i)T + 59iT^{2} \)
61 \( 1 + (-6.94 - 6.94i)T + 61iT^{2} \)
67 \( 1 + (-7.89 - 7.89i)T + 67iT^{2} \)
71 \( 1 - 12.6T + 71T^{2} \)
73 \( 1 + (-4.83 + 4.83i)T - 73iT^{2} \)
79 \( 1 + (5.31 + 5.31i)T + 79iT^{2} \)
83 \( 1 + (-8.43 - 8.43i)T + 83iT^{2} \)
89 \( 1 + (11.6 - 11.6i)T - 89iT^{2} \)
97 \( 1 + 5.04iT - 97T^{2} \)
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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

−11.29573403388648904768073967404, −10.93440365544469145184947695348, −9.937847353555167596249882138372, −9.349152029015375479298843832931, −8.436887787965877128561640752249, −6.86059249979427211175617853435, −6.00909457227823162922520337482, −5.46584747974478922396777299584, −3.53985366745794822970880741945, −2.31616812252778997381831312401, 0.43459862122650478939976492554, 1.78589529398794047180456442993, 3.74699968918432392976650693774, 5.36693285188112948028517390926, 6.51447615498759902722389256284, 6.89367623060756308030117597941, 8.087816504146326162100051505116, 9.391317234961806335481965446352, 9.903262935357367270674746772401, 10.81499756231601028421522873400

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