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Properties

Label 2-975-195.8-c0-0-0
Degree $2$
Conductor $975$
Sign $0.966 - 0.256i$
Analytic cond. $0.486588$
Root an. cond. $0.697558$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 − 0.707i)3-s + 4-s + 1.41i·7-s + 1.00i·9-s + (−0.707 − 0.707i)12-s + (0.707 + 0.707i)13-s + 16-s + (−1 + i)19-s + (1.00 − 1.00i)21-s + (0.707 − 0.707i)27-s + 1.41i·28-s + (−1 − i)31-s + 1.00i·36-s − 1.41i·37-s − 1.00i·39-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)3-s + 4-s + 1.41i·7-s + 1.00i·9-s + (−0.707 − 0.707i)12-s + (0.707 + 0.707i)13-s + 16-s + (−1 + i)19-s + (1.00 − 1.00i)21-s + (0.707 − 0.707i)27-s + 1.41i·28-s + (−1 − i)31-s + 1.00i·36-s − 1.41i·37-s − 1.00i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 - 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 - 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $0.966 - 0.256i$
Analytic conductor: \(0.486588\)
Root analytic conductor: \(0.697558\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :0),\ 0.966 - 0.256i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.016330664\)
\(L(\frac12)\) \(\approx\) \(1.016330664\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 \)
13 \( 1 + (-0.707 - 0.707i)T \)
good2 \( 1 - T^{2} \)
7 \( 1 - 1.41iT - T^{2} \)
11 \( 1 + iT^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 + (1 - i)T - iT^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + (1 + i)T + iT^{2} \)
37 \( 1 + 1.41iT - T^{2} \)
41 \( 1 - iT^{2} \)
43 \( 1 + (-1.41 + 1.41i)T - iT^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 - iT^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + 1.41T + T^{2} \)
71 \( 1 - iT^{2} \)
73 \( 1 - 1.41T + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 - iT^{2} \)
97 \( 1 + 1.41T + T^{2} \)
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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

−10.65221155762070414517948967213, −9.337583858319189960235091962548, −8.459688268653417083171588633559, −7.61080779888646819442563109291, −6.71514035016707881837909317654, −5.89070990282977991581785212141, −5.59127050424141662676893264092, −3.96917881110752077191163361478, −2.40746797908336156898807806421, −1.79443879774017476328636214715, 1.16407620473532112688886912447, 3.00639951963409543631145218188, 3.93653368868660025731766528605, 4.88698430589032533651551713885, 6.02477256201757938796492405637, 6.69601581289193307198546992262, 7.44326526704159022067054932995, 8.488601605486281541362511835356, 9.637813428142605537624023915242, 10.55246638158536707821181062055

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