Properties

Label 2-1205-5.4-c1-0-74
Degree $2$
Conductor $1205$
Sign $-0.656 + 0.754i$
Analytic cond. $9.62197$
Root an. cond. $3.10193$
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  − 1.83i·2-s + 2.40i·3-s − 1.36·4-s + (−1.46 + 1.68i)5-s + 4.41·6-s − 0.302i·7-s − 1.17i·8-s − 2.78·9-s + (3.09 + 2.69i)10-s − 5.09·11-s − 3.27i·12-s − 4.63i·13-s − 0.554·14-s + (−4.05 − 3.53i)15-s − 4.87·16-s − 1.03i·17-s + ⋯
L(s)  = 1  − 1.29i·2-s + 1.38i·3-s − 0.680·4-s + (−0.656 + 0.754i)5-s + 1.80·6-s − 0.114i·7-s − 0.414i·8-s − 0.929·9-s + (0.977 + 0.850i)10-s − 1.53·11-s − 0.944i·12-s − 1.28i·13-s − 0.148·14-s + (−1.04 − 0.912i)15-s − 1.21·16-s − 0.250i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.656 + 0.754i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.656 + 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1205\)    =    \(5 \cdot 241\)
Sign: $-0.656 + 0.754i$
Analytic conductor: \(9.62197\)
Root analytic conductor: \(3.10193\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1205} (724, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1205,\ (\ :1/2),\ -0.656 + 0.754i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7789638867\)
\(L(\frac12)\) \(\approx\) \(0.7789638867\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.46 - 1.68i)T \)
241 \( 1 - T \)
good2 \( 1 + 1.83iT - 2T^{2} \)
3 \( 1 - 2.40iT - 3T^{2} \)
7 \( 1 + 0.302iT - 7T^{2} \)
11 \( 1 + 5.09T + 11T^{2} \)
13 \( 1 + 4.63iT - 13T^{2} \)
17 \( 1 + 1.03iT - 17T^{2} \)
19 \( 1 - 8.55T + 19T^{2} \)
23 \( 1 + 1.34iT - 23T^{2} \)
29 \( 1 + 10.3T + 29T^{2} \)
31 \( 1 - 0.459T + 31T^{2} \)
37 \( 1 + 1.55iT - 37T^{2} \)
41 \( 1 - 5.44T + 41T^{2} \)
43 \( 1 + 5.45iT - 43T^{2} \)
47 \( 1 + 10.3iT - 47T^{2} \)
53 \( 1 + 4.20iT - 53T^{2} \)
59 \( 1 - 5.01T + 59T^{2} \)
61 \( 1 + 6.46T + 61T^{2} \)
67 \( 1 + 14.8iT - 67T^{2} \)
71 \( 1 + 6.08T + 71T^{2} \)
73 \( 1 + 13.8iT - 73T^{2} \)
79 \( 1 + 2.63T + 79T^{2} \)
83 \( 1 - 2.13iT - 83T^{2} \)
89 \( 1 - 2.74T + 89T^{2} \)
97 \( 1 - 10.5iT - 97T^{2} \)
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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

−9.866464851000724110158940321053, −9.101169771790301719635045382816, −7.80018649118057731614642681678, −7.28880789878092993538842445235, −5.60805060251477763575068973498, −4.94277962966098276772253882086, −3.71487045239652882991051036454, −3.30903345083082092928303447102, −2.47537626910827574215135460267, −0.33752471250844118485369141109, 1.39161902117186887677901401909, 2.66950083451573109470137651518, 4.31563684294715123306739004132, 5.39763817505646393821277099011, 5.87376097511247752454622340017, 7.11750127687440298765766915572, 7.49232853858509430394166394485, 7.928878931673530549605147608410, 8.816845977785333491987942596513, 9.608912479670940968658617810381

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