Properties

Label 1-373-373.107-r0-0-0
Degree $1$
Conductor $373$
Sign $-0.944 + 0.327i$
Analytic cond. $1.73220$
Root an. cond. $1.73220$
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  + (0.780 + 0.625i)2-s + (−0.839 − 0.543i)3-s + (0.217 + 0.975i)4-s + (0.470 + 0.882i)5-s + (−0.315 − 0.948i)6-s + (−0.250 + 0.968i)7-s + (−0.440 + 0.897i)8-s + (0.409 + 0.912i)9-s + (−0.184 + 0.982i)10-s + (−0.557 + 0.830i)11-s + (0.347 − 0.937i)12-s + (−0.440 − 0.897i)13-s + (−0.801 + 0.598i)14-s + (0.0843 − 0.996i)15-s + (−0.905 + 0.425i)16-s + (−0.954 − 0.299i)17-s + ⋯
L(s)  = 1  + (0.780 + 0.625i)2-s + (−0.839 − 0.543i)3-s + (0.217 + 0.975i)4-s + (0.470 + 0.882i)5-s + (−0.315 − 0.948i)6-s + (−0.250 + 0.968i)7-s + (−0.440 + 0.897i)8-s + (0.409 + 0.912i)9-s + (−0.184 + 0.982i)10-s + (−0.557 + 0.830i)11-s + (0.347 − 0.937i)12-s + (−0.440 − 0.897i)13-s + (−0.801 + 0.598i)14-s + (0.0843 − 0.996i)15-s + (−0.905 + 0.425i)16-s + (−0.954 − 0.299i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.944 + 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.944 + 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(373\)
Sign: $-0.944 + 0.327i$
Analytic conductor: \(1.73220\)
Root analytic conductor: \(1.73220\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{373} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 373,\ (0:\ ),\ -0.944 + 0.327i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1935483120 + 1.148113198i\)
\(L(\frac12)\) \(\approx\) \(0.1935483120 + 1.148113198i\)
\(L(1)\) \(\approx\) \(0.8692763189 + 0.6956680852i\)
\(L(1)\) \(\approx\) \(0.8692763189 + 0.6956680852i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad373 \( 1 \)
good2 \( 1 + (0.780 + 0.625i)T \)
3 \( 1 + (-0.839 - 0.543i)T \)
5 \( 1 + (0.470 + 0.882i)T \)
7 \( 1 + (-0.250 + 0.968i)T \)
11 \( 1 + (-0.557 + 0.830i)T \)
13 \( 1 + (-0.440 - 0.897i)T \)
17 \( 1 + (-0.954 - 0.299i)T \)
19 \( 1 + (0.820 - 0.571i)T \)
23 \( 1 + (-0.0506 - 0.998i)T \)
29 \( 1 + (-0.972 - 0.234i)T \)
31 \( 1 + (0.347 + 0.937i)T \)
37 \( 1 + (-0.378 + 0.925i)T \)
41 \( 1 + (0.979 + 0.201i)T \)
43 \( 1 + (0.217 + 0.975i)T \)
47 \( 1 + (-0.117 + 0.993i)T \)
53 \( 1 + (0.409 - 0.912i)T \)
59 \( 1 + (0.283 + 0.959i)T \)
61 \( 1 + (-0.839 - 0.543i)T \)
67 \( 1 + (0.528 - 0.848i)T \)
71 \( 1 + (0.736 + 0.676i)T \)
73 \( 1 + (0.585 + 0.810i)T \)
79 \( 1 + (-0.184 + 0.982i)T \)
83 \( 1 + (0.409 - 0.912i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (-0.954 - 0.299i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−24.050410626773839276254378501350, −23.31833039936585356659097751353, −22.32738346448789108883642895987, −21.607057288368694497668505110844, −20.89827371208052666060817666158, −20.191523181306680498415115155064, −19.17818612244860214464149649415, −17.98036416067891148066264826040, −16.9133801078062640948861504822, −16.28120559283756872833096111000, −15.43120234488760512463228943023, −14.0298100554165535361480103748, −13.40716905970739573122553878455, −12.50724375632732127447803842094, −11.52121481294169678843353221102, −10.769473288657500824144015536321, −9.824756882500979563721798784422, −9.12770235303560415238586903623, −7.262469754175099953575118139, −6.03996905429122214451598062809, −5.361553772566634833121566501240, −4.343962003824514942572222771435, −3.62672831523469507639712747171, −1.87289446677914622261946668072, −0.571741087400039810933654706926, 2.25192066687619443845266530978, 2.885236304410749539676327285005, 4.75529008893394540139025652280, 5.465800381573810755749290502324, 6.39784392905961098372699217921, 7.09036835976496230018870727078, 8.03249012212341564667966136502, 9.5545011956419094135883068245, 10.76953402030567527270530008961, 11.67158904124808301268739322209, 12.65256314628536285810024996384, 13.17291739143652126791935234090, 14.31922830719424400003348207122, 15.334247405328918287303039522507, 15.85621683821960600200943367218, 17.16386528510411481034196852178, 18.02095504292212997453477915144, 18.26978957065398411248352266085, 19.727521127829255977420708698661, 21.01246981288463494011538006045, 22.00431206392918456134880577197, 22.6036494957331470706620638098, 22.87233220940992651143350737759, 24.339190633152800351520515857227, 24.70093548756021043704033088950

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