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Properties

Label 2-975-195.122-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: $\chi_{975} (707, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :0),\ 0.966 + 0.256i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.493721346\)
\(L(\frac12)\) \(\approx\) \(1.493721346\)
\(L(1)\) 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

−10.11149143514414099900515386535, −8.976458039786263590569938632109, −8.709886446932961134742320135830, −7.51343590167377160791255212441, −6.87436911556631326359461900205, −6.13537524279281411465639013091, −5.11642301443041347037058423047, −3.53614776203931664106644489193, −2.40158454366151255387916609791, −1.98773887217548407872535815190, 1.77433405545193753888318051829, 3.00982193508563821022170026885, 3.84603696817627017961781432979, 4.81135546985067423365053977176, 6.02506133524096343639942600692, 7.07274232084790428169883857894, 7.77324619798820223701139872304, 8.354562475456836247071094615003, 9.821244410293337256236059670966, 10.14683842110656372928295187848

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