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Properties

Label 2-975-195.164-c1-0-37
Degree $2$
Conductor $975$
Sign $-0.576 - 0.816i$
Analytic cond. $7.78541$
Root an. cond. $2.79023$
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  + (1.90 + 1.90i)2-s + (1.66 − 0.460i)3-s + 5.29i·4-s + (4.06 + 2.30i)6-s + (−1.58 − 1.58i)7-s + (−6.28 + 6.28i)8-s + (2.57 − 1.53i)9-s + (3.68 + 3.68i)11-s + (2.43 + 8.83i)12-s + (−1.62 + 3.21i)13-s − 6.05i·14-s − 13.4·16-s − 1.75i·17-s + (7.85 + 1.98i)18-s + (0.812 + 0.812i)19-s + ⋯
L(s)  = 1  + (1.35 + 1.35i)2-s + (0.964 − 0.265i)3-s + 2.64i·4-s + (1.66 + 0.942i)6-s + (−0.599 − 0.599i)7-s + (−2.22 + 2.22i)8-s + (0.858 − 0.512i)9-s + (1.11 + 1.11i)11-s + (0.703 + 2.55i)12-s + (−0.450 + 0.892i)13-s − 1.61i·14-s − 3.35·16-s − 0.426i·17-s + (1.85 + 0.467i)18-s + (0.186 + 0.186i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.576 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.576 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $-0.576 - 0.816i$
Analytic conductor: \(7.78541\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (749, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ -0.576 - 0.816i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.94267 + 3.75041i\)
\(L(\frac12)\) \(\approx\) \(1.94267 + 3.75041i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.66 + 0.460i)T \)
5 \( 1 \)
13 \( 1 + (1.62 - 3.21i)T \)
good2 \( 1 + (-1.90 - 1.90i)T + 2iT^{2} \)
7 \( 1 + (1.58 + 1.58i)T + 7iT^{2} \)
11 \( 1 + (-3.68 - 3.68i)T + 11iT^{2} \)
17 \( 1 + 1.75iT - 17T^{2} \)
19 \( 1 + (-0.812 - 0.812i)T + 19iT^{2} \)
23 \( 1 + 4.74iT - 23T^{2} \)
29 \( 1 + 4.37iT - 29T^{2} \)
31 \( 1 + (0.454 + 0.454i)T + 31iT^{2} \)
37 \( 1 + (-0.906 - 0.906i)T + 37iT^{2} \)
41 \( 1 + (2.81 - 2.81i)T - 41iT^{2} \)
43 \( 1 + 3.01T + 43T^{2} \)
47 \( 1 + (-2.05 + 2.05i)T - 47iT^{2} \)
53 \( 1 + 8.31T + 53T^{2} \)
59 \( 1 + (0.222 + 0.222i)T + 59iT^{2} \)
61 \( 1 - 9.69T + 61T^{2} \)
67 \( 1 + (-10.0 + 10.0i)T - 67iT^{2} \)
71 \( 1 + (-8.02 + 8.02i)T - 71iT^{2} \)
73 \( 1 + (0.537 + 0.537i)T + 73iT^{2} \)
79 \( 1 + 8.21T + 79T^{2} \)
83 \( 1 + (-0.470 - 0.470i)T + 83iT^{2} \)
89 \( 1 + (-4.20 - 4.20i)T + 89iT^{2} \)
97 \( 1 + (-5.84 + 5.84i)T - 97iT^{2} \)
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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

−9.879107578014133655205218890324, −9.261813475824204578876222812929, −8.272440891538829160029187535025, −7.43947376261790203534214888253, −6.73734220882469196915189900769, −6.46799493067815712789219375233, −4.84626131985768299300465362334, −4.18210099299611425519707483233, −3.47097742887402463868692409832, −2.22629931375187930139495640223, 1.30955485055238032858294798794, 2.58866753708888077080782354396, 3.34750339272053548933136050229, 3.88220881087583682622312773713, 5.13204399261856257504435169925, 5.86349885166005384562343253243, 6.89330346521918666767998250661, 8.391653016196738888593442490501, 9.280054175858580385871851208983, 9.799786639986454246392274521795

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